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QUADRIC REPRESENTATION
7WD SUBMANIFOLDS OF PIA/(TE, TYPE;

presented by

IVKO DtMITRté

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity Institution

 

 

QUADRIC REPRESENTATION AND
SUBMANIFOLDS OF FINITE TYPE

By

Ivko Dimitrié

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1989

All/RV N.m AvhJ nJ

téU

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a

bOUl

ABSTRACT
QUADRIC REPRESENTATION AND
SUBMANIFOLDS 0F FINITE TYPE
By
Ivko Dimitric

For an isometric immersion x : Mn --9 Em of a Riemannian manifold into a
Euclidean space, one defines the map it” = x-xt ( x regarded as column vector) from M into
the set of m x m symmetric matrices, which we call quadric representation of M and
propose to study it .

A smooth map f : Mn —-) Em is said to be of finite type (k-type) if it can be
decomposed into finitely many ( k, not counting constant vector) eigenvectors of the
Laplacian . In particular, a manifold immersed into a Euclidean space is said to be of
k-type if the corresponding immersion is of k - type.

We prove some general results about the quadric representation, in particular those
related to the condition of if being of finite type. Submanifolds for which i’ is l-type map
are classified as totally geodesic spherical submanifolds. We show that for minimal
submanifold of Em the quadric representation is of infinite type. Further, we classify
compact spherical hypersurfaces which are of 2-type via 51' as small hyperspheres or
standard products Sp(rl) x S"’9(r2) with only three different possibilities for (r1, r2). The
main result is classification of compact minimal spherical hypersurfaces which are of 3-type
and mass-symmetric via 56 in dimensions n S 5 . The only such submanifold is the Cartan
hypersurface SO(3)/Z2 x 2.2 . At the end we begin the study of submanifolds of Em whose
mean curvature vector is harmonic. Such submanifolds are shown to be minimal under

additional assumptions (e.g. for hypersurfaces having at most two distinct principal

curvatures).

To my mother Nadeida ,
brother Radoslav,

and in memory of my father Milan

iii

 

ACKNOWLEDGMENTS

I wish to express my deep gratitude to Professor Bang-yen Chen, under whose
expert and patient guidance was this work done. He provided not only help on the topic but
also valuable insights into Differential geometry as a whole. I would like to express my
appreciation to Professors Blair and Ludden for their helpful conversations and sharing
their knowledge in preparing this subject. In addition, thanks go to all my teachers at
Michigan State University for their excellent teaching. Finally, I especially thank my

mother and brother for their love and encouragement.

iv

INTRODUCTION

CHAPTER 1.

CHAPTER 2.

CHAPTER 3.

CHAPTER 4.

SUMMARY

BIBLIOGRAPHY

TABLE OF CONTENTS

Preliminaries
1. Riemannian geometry and submanifolds
2. Homogeneous spaces
3. Second standard immersion of a sphere
4. Isoparametric spherical hypersurfaces

5. Finite type maps and submanifolds
Quadric representation of a submanifold

Spherical hypersurfaces which are of low type
via the second standard immersion of the sphere

1. Spherical hypersurfaces which are of 2 - type via a:

2. Minimal spherical hypersurfaces which are of
3-typeandmass-symmetricvia 32'

Submanifolds of Em with harmonic mean curvature vector

13
17
21
28

34

53

53

83

96

100

INTRODUCTION

For an isometric immersion x : Mn —-) Em of a smooth Riemannian manifold Mn
into a Euclidean space, one of the first questions one might ask is : " What are the natural
maps related to the immersion x ? " Of course, x itself is one such map and we have rich
submanifold theory of isometric immersions . Another natural map is the Gauss map which
corresponds to each point p of M , the tangent space of M at p , p —-) e1(p) A A en(p),
and investigation of this map led to many interesting results. Then each vector field
X e I‘(TM) on M defines a map X : Mn -) Em and the Hopf index theory handles one
aspect of this map . Also, if we regard x as a column matrix, x(p) = (X1(P). , xm(p))t ,
then one defines a map 32' from M into the set of m x m symmetric matrices ( which is also
a Euclidean space) by i' = x-x‘. We call this map the quadric representation of M and
propose to study it . The map if is not necessarily an isometric immersion but if i is
assumed to be isometric (or just conformal ) it follows that M must be a submanifold of a
sphere centened at the origin (Theorems 2.1-2) .

There are several important results about integrals of geometric quantities on a
compact Riemannian manifold M . The classical theorem of Gauss - Bonnet states that
IKdV = 2n x(M) , i.e. the integral of the Gauss curvature is a topological invariant - the
Euler characteristic. Also the celebrated inequality of Chem and Lashof gives a universal
lower bound (topological) for so called total absolute curvature: TA(x) 2 b(M) , where
b(M) is the total Betti number of M [Ch—L]. Up to late 1970's there were some indications

that one could find esrimates for the total mean curvature in terms of the Riemannian

structure of M. Finally, in 1979, BY. Chen gave the following best possible estimate for

the total mean curvature [C 2]

n

5" vol(M) s IIHIZdv s 5: vol(M) ,
M

where 21, and 71.9 are two eigenvalues of the Laplacian uniquely determined by the spectral
behavior of the immersion x . Thus we get an invariant [p, q] associated with M where p is
an integer 2 1 and q is either an integer 2 p or no (in latter case right hand side of the
inequality is co ). A submanifold M (or an immersion x ) is said to be of finite type if q is
finite. Equivalently, M is of finite type if the immersion x decomposes into finitely many

eigenvectors of the Laplacian ,

x = x0+ xp+...+ xq , where x0=const and Axt= 1.x, forallpStSq.

If M is compact, the constant vector x0 is the center of mass of M . A submanifold M is of
k - type if there are exactly k nonzero vectors xt (t > 0) in the decomposition above. The
same definition can be adopted if we do not assume M compact, and also if x is not
necessarily an isometric immersion but simply an arbitrary smooth map from M into Em .
Since its inception, the theory of finite type submanifolds has become an area of active
research [C 4]. According to the well known theorem of Takahashi [Ta 1], compact 1 -
type submanifolds of E"1 are characterized as being minimal in hypersphere and one can
expect that 2 - type and higher type submanifolds are more general. Indeed, the
classification of even 2 - type spherical submanifolds is virtually impossible, but finite type
submanifolds are still "nice" examples of submanifolds.

In Chapter 2 we classify submanifolds x : M“ -9 Em for which the quadric
representation i is of 1 - type as totally geodesic submanifolds of hypersphere of E“.
While it is relatively easy to construct nonspherical submanifolds for which a: is of finite

type, we show that if M is minimal in Em than its quadric representation is of infinite type

(Theorem 2.4) . Next, in Chapter 3 we study spherical hypersurfaces which are of low
type via quadric representation. Studying submanifolds x : Mn —) Em whose quadric
representation is of finite type amounts to studying spectral behavior of products of
coordinate functions xi-xj . We classify spherical hypersurfaces which are of 2 - type via i
as products of two spheres with three different possibilities for the radii, thus generalizing
a result of M. Barros and B.Y. Chen [B-C] . Investigation of 3 - type spherical
submanifold is much more complicated because of the computation of iterated Laplacians
involved. The only known result about spherical submanifolds being of 3 — type via 2' is
classification of minimal surfaces (n = 2) in sphere which are of 3 - type by M. Barros and
F. Urbano [B-U] (See also [U]). In Chapter 3 we undertake study of minimal
hypersurfaces of sphere which are mass - symmetric and of 3 - type via a: . The only such
submanifold in dimensions n S 5 is the Cartan hypersurface SO(3)/Z2sz (Theorem
3.2.2). Actually, all minimal isoparametric spherical hypersurfaces with three distinct
principal curvatures are also mass - symmetric and of 3 - type via 32' (Lemma 3.2.3).

In Chapter 4 we study submanifolds x : Mn —9 13m of a Euclidean space which
satisfy AH == 0 , where H is the mean curvature vector of the immersion. This condition
is equivalent to Azx = 0 . Minimal submanifolds being the trivial solution, the real
problem is to find nonminimal examples, that is, those immersions which are biharmonic
but not harmonic. While the construction of such examples (if they exist) seems difficult,
we show that submanifolds satisfying AH = 0 are necessarily minimal if any of the
following conditions is satisfied

(1) M11 has constant mean curvature .

(2) M“ is a hypersurface of BMI with at most two distinct principal curvatures .

(3) M“ is a pseudoumbilical submanifold of E“ (n at 4)
(4) M“ is of finite type.

CHAPTER 1
PRELIMINARIES

The purpose of this introductory chapter is to supply necessary definitions and to
outline ideas and some general techniques used in the subsequent chapters. We deem it
good to have main facts that will be used assembled in one place for easy reference without
having to digress from the main flow later. This overview is by no means supposed to be
exhaustive, but rather to assist a potential reader in reading through the rest of the work
without necessity to turn to the references frequently. Most of the material in this chapter,

however, is well known.
1. Riemannian geometry and submanifolds

Standard references here are [K-N] and [C 1] . We assume elementary notions from
the theory of differentiable manifolds (differentiable functions, vector fields, tensor and
exterior algebras, connections, integration on compact manifolds, ...) known. All
manifolds are real, and (with the possible exception of some Lie groups) will be assumed
connected. A generic manifold is usually denoted by M", where it stands for the
dimension, or simply by M. The word "differentiable" means "C°°- differentiable" and is
synonymous with "smooth“. All manifolds and geometric objects will be assumed smooth
unless stated otherwise. The set of real-valued smooth functions on M is denoted by

C°°(M), and the algebra of differentiable functions in the neighborhood of p by C°l;’(M).

dimension. A tangent vector X to a manifold M at a point p e M is a linear map from

C?(M) to R, which is a derivation of the algebra C";(M) , that is
(1.1.1) X(fg)=(Xf)g+f(Xg) , forevery f,ge C°’§’(M) .

The set of all tangent vectors at p, with its natural vector space structure, is called the
tangent space of M at p and is denoted by TpM. It can be visualized as the set of tangent

vectors at p to all curves in M passing through p. The set of all pairs (p,TpM) forms the
tangent bundle TM which is a vector bundle over M. A smooth section of TM is just a
vector field on M, and the set of those is denoted by 1‘ (TM). For two vector fields X, Y,
the bracket [X,Y] is the vector field defined as

(1.1.2) [X,Y] f=X(Yf) - Y(XO .

For every function f e C°°(M) we can define l-form df, called the difi'erential of
f, by df(X) = Xf , for every f e 1‘ (TM) . More generaly, for a map f : M -—) N between
two manifolds and a point p e M we have the induced map (fa‘)p : TPM —-> Tf®)N , called
difi'erential of f at p , defined as

(f*(X))g=X(g-f) , forevery gECf$)(N) and XeTpM.

The Pull-back map at (at f(p)) is the adjoint of this linear map.
An (afl‘ine) connection on M is a rule V which assigns to each vector field X a

linear map VX of the vector space FCIM) into itself satisfying the following two conditions

VtX+gY = fo +gVY

(1.1.3)
Vx(fY) = foY+(Xf)Y

 

extended to arbitrary tensor fields in a natural way to produce derivation of the tensor

algebra that commutes with contractions , e. g. for covariant 2-tensor T we have

(pror. Z) = meYz» - T< VxY. Z) - m. we .
Given a coordinate neighborhood (U, x1, , x") of a manifold Mn , we have the
coordinate vector fields 81 = i 8 = —- on U. In the presence of a connection

8x1 ’ ’ " 8x“
V , we can define functions I"; called the Christofi'el symbols by

(1.1.4) V8531) = z: rigs]t

Let y: I -9 M be a curve in M. The tangent vector field to the curve, T(t) = 74%) , is
called the velocity vector field of the curve 7 . The curve 7 is called a geodesic (of a
connection V) if VTT = 0 , i.e. the velocity vector field is parallel along the curve. Using

the affine connection Von M we define two tensor fields, curvature tensor R and torsion

tensor T by
(1.1.5) R(X,Y) = VXVY - VYvX — V[X,Y]
(1.1.6) T(X,Y) = VXY — VYx — [X,Y] , X, Ye T(TM)

A Riemannian manifold (M,g) is a differentiable manifold M equipped with a symmetric
positive definite tensor field g of type (0,2), called the Riemannian metric. On a

Riemannian manifold there exists a unique affine connection V which has zero torsion,

T E 0 , and such that the metric tensor is parallel, Vg = O . These two conditions are

equivalent to

(1.1.7) [X,Y] = VXY - VYX

(1.1.8) 2 g(st) = g(vzx 9 Y) + g(x 9 VZY) s

for every vector fields X, Y, Z . This connection is called the Levi-Civita (or
Riemannian) connection. The Christoffel symbols of this connection are computed in a

local coordinate system (U, x1, , x“) as

(1.1.9) fir;- 2gtk 5%! + 35-: - 35:21]
As usual, (gij) denotes matrix of the metric tensor g and (gij) is its inverse matrix.

For each point p e M and each 2-plane II c TpM , the sectional curvature K(II)
of II is defined by K(II) = g(R(X,Y)Y, X) , where X, Y are orthonormal vectors which
span II (it is independent of the choice of such pair X,Y in II). Given two vectors X and Y
in TpM and an orthonormal basis e1, , e“ of TpM we define the Ricci tensor 8 and

the scalar curvature 1: at p by

(1.1.10) S(X,Y) = 2g(R(x,e,)e, , Y)
1

(1.1.11) 1 = 25(Ci.61)
i

If for a Riemannian manifold (M, g) the sectional curvature K(II) is constant for all planes
II c: TpM and all points p e M , then M is called a space of constant curvature or a
space form. Standard examples are : Euclidean space Em (sectional curvature is 0), Sphere
Sm(r) (curvature is 1/r2 > O ), and hyperbolic space Hm (curvature < 0) . Under additional
topological assumptions (complemess, simply connectedness) these are the only ones. A

manifold (M,g) is called (locally) flat if its sectional curvature is 0.

A map f : (M, g) —-> (N, h) between two Riemannian manifolds is called conformal
if f*h = (p g for some positive function e on M . If 4’ is a positive constant f is
homothetic . If 4) a l and f is a diffeomorphism then f is called an isometry. (M,g) is
called conformallyflat if there is a metric on M conformal to g with respect to which M is
flat.

Let (M, g) and (N, h) be two Riemannian manifolds. Then one can define a

Riemannian metric g x h on the product manifold M x N in the following way
(g x h )(X. Y) = g (X1. Y1) + h (X2. Y2) .

where X = XI + X2 and Y = Y1 + Y2 are the decompositions of X and Y with respect
to the sum T(mm)(M x N) a TmM O TnN .
Given a Riemannian manifold (M, g) and a point p e M . For each vector X in
TM there is a unique geodesic 'yx(t) defined in the neighborhood of 0 such that yx(0) = p
and 7,}(0) = X . We define expr as the point in M given by 7x0) when yx(1) is
defined. The map expp is called the exponential map at p. For each p e M , there is an
open neighborhood U of 0 6 TM and an Open neighborhood U of p e M such that the
exponential map expp: U -> U is a diffeomorphism of U onto U. Let U and U be as
above, and let e1, , el1 be an orthonormal basis of TPM . For each X e U we put
X = xlel + + x“en . Then the components x1 , , x“ are called normal coordinates
of the point q = expr in U (determined by the frame e1, , en ). In the normal
coordinate system (U, x1 , , xn ) we have gij(p) = Sij and I": (p) = 0, i.e. Vei°j(P) = O
for every i, j, k. A Riemannian metric is called complete if every geodesic can be extended
indefinitely in both directions, equivalently, if expr is defined for every point p and
every vector X 6 TM .This corresponds to the topological completness of the metric
space M , where the distance between two points is defined as the infimum of the lengths

of curves joining the two points. Every compact Riemannian manifold is complete.

9

A map x : M -) M is called an immersion if (X*)p : TpM -> Tme is injective
for each p e M. If, in addition, x itself is injective it is called an embedding . If (M, g) and
(M, E ) are both Riemannian manifolds , x is an isometric immersion if x*'g' = g . When
this is the case we say that the metric on M is induced from that of M, and call M
submanifold of hi . We shall identify x with its image x*(X) for any x 6 TM.
Corresponding to the orthogonal splitting

(1.1.12) TpM = TM 6 T-IgM , forevery pe M

we can write for (local) smooth vector fields X and Y on M

N

(1.1.13) Vx...x x*Y .-= x,(va) + h(X, Y) ,

where VxY tangent to M and h (X, Y) is normal to M. Note that in general symbols with ~
denote objects on M and without ~ objects on M. According to the convention above we
will also supress writing x* in the sequel. We call V the induced connection of M (it is
actually the Levi-Civita connection of (M, g) ), and normal bundle valued symmetric tensor
field h we call the second fundamental form of the immersion. If h -:-: 0 , the submanifold
M is called totally geodesic . An immersion x is said to be full if x(M) does not lie in any
totally geodesic submanifold of M . Let g be a local normal vector field and X a vector
field on M then we have the following orthogonal decomposition

(1.1.14) fix: = —A§X + Dxé .

where - A§X and Dxfi are the tangential and normal components of Vxé respectively.
For every é , Ag is an endomorphism of tangent space of M at every point. It is known as
the Weingarten map or shape operator of§ and is related to the second fundamental

form h via

 

10

(1.1.15) toluene = g(A§X.Y) .

A§X is a symmetric operator and as such can be diagonalized over the reals. Its
eigenvectors are called principal directions of g and its eigenvalues, principal
curvatures.

Let e1, . en . en“, , em be an adapted frame, i.e. local frame of ortho-
normal vector fields of M along M such that the first 11 vectors are tangent to M and the

remaining m - n are normal to M . We adopt the following convention about the range of

indices:1$i,j,k,...$n , n+15r,s,...Sm and 15A,B,C,...Sm .We

define a normal vector field H by
1 n 1 m
(1.1.16) H = fl h(ei,ei) = ii 2 (tr Ar)er
i=1 n=n+l

and call it the mean curvature vector field . A submanifold M (or an immersion x ) is
called minimal if H a O . If we choose em] to be in the direction of H, °n+1 H H , then
H = elem] for some real function or which is called the mean curvature of M. If A: = p I
for some function p, we say that é is an umbilical section. If every local normal section is
umbilical, submanifold is called (totally) umbilical. Equivalently, a totally umbilical
submanifold is characterized by the property h(X, Y) = g(X, Y) H , for every
X,Y e I‘(TM) . A submanifold is called pseudoumbilical if AH = p I . It is called

quasiumbilical if there exists an orthonormal frame of local normal vector fields

e , em 6 T-LM such that for every r , all principal curvatures of er , except

n+1"“
possibly one, are equal.
The normal part of (1.1.14 ) , D , defines a metric connection in the normal bundle

T‘LM i.e. Dx( §(§,n)) =-. g (ox: , n) + g(g, Dxn) . Its curvature will be denoted by RD.

Let e1, , em be a local orthonormal frame of vector fields defined on an open

 

11

set U of a Riemannian manifold M” . Denote by (01, , com the dual frame, and

define m2 connection 1 -forms (of: on U by

m
(1.1.17) vch = E df(an.
-1

Then mg + mgt = 0 , and the following structural equations of Cartan hold

(1.1.18) dtoA = —Z to; a to”
(1.1.19) deg = ~20); mg + 523.

where (23 =é- Z figco (DCA (0D with RSCD = g ( R(ec, eD) eB, eA) . In the space of

constant curvature, Mm(c) , we have 93 = c (0“ A (DB .

Now if Mn is a submanifold of Mm and e1, , en, en +1, , em an adapted
frame, then when the forms to: are restricted to M we see that to; are connection l-fonns

of the induced connection V, a); are connection 1-forrns of the normal connection D , and

mi determine the second fundamental form h. Moreover, by a lemma of Cartan

(1.1.20) tot = thmi , where hi‘j=‘g‘(h(ei,ej),er).

Let x : Mn —t Mm be an isometric immersion . Then the three fundamental

equations of Gauss, Codazzi and Ricci "determine" immersion x (cf. [C 4], p. 120). For the

immersion into a space of constant curvature c , x : Mn -—) Mm(c) , equations of Gauss,

Codazzi and Ricci are respectively given by

(1.1.21) R(X.Y;Z.W) = Cl g(X,W)g(Y,Z) - g(X,Z)g(Y,W)}

 

12

1' §' (h(X.W). h(Y. 2)) - E (h(X. Z). 110’. W»

(1.1.22) (vxth, 2)

(1.1.23) RD(X. Y; §. 11)

g([A§,An]X.Y)
Here, V is so called connection of van der Waerden - Bortolotti defined by

(1.1.24) (Vxh)(Y,Z) = thor, Z) - h(VxY,Z) - h(Y,VXZ) ,

and K(X. Y; z. W) = g( R(X. Y) z . W) . 12%. Y; t. n) = z (RD(X. Y) i. 11).
If M is a hypersurface of space of constant curvature c we have only Gauss and Codazzi

equations which in this case read as
(1.1.25) R(X, Y) = C(XAY) + AXAAY
(1.1.26) (VXA)Y == (VyA)X

If e1, , en is orthonormal basis of principal directions of A , ll, , kn respective

principal curvatures and to" corresponding connection forms, then the Codazzi equation is

equivalent to the following system of formulas
(1.1.27) (tj - 7t,)m}(c,) = ejx, , i¢ j

(1.1.28) (7tj - 1k) a)§(e,) = (2.,- 1k)m‘i‘(ej) , i¢j¢k¢i

and no summation occurs on repeated indices .

 

13

2. Homogeneous spaces

For the basic facts about Lie groups we refer to [W a], [He] and for homogeneous
spaces to [K-N], [Ch-E], [Ch 1], [Be] .

A Lie group G is a smooth manifold (which we do not assume connected), which
has the structure of a group in such a way that the map 4) : G x G -) G defined by
(h(X, y) --= x- y'1 is smooth. The identity component of a Lie group is itself a Lie group.
Readily available examples of Lie groups are classical groups GL(n), 0(n), SO(n), U(n),
Sp(n), etc. Also, the well known result of Myers and Steenrod asserts that the isometry
group of any Riemannian manifold is a Lie group.

A Lie algebra over R is a real vector space V together with a bilinear map (called

bracket) [, ]:VxV—) V suchthatforany x,y,ze V

(1.2.1) (X. y] = - [y. x}

(1.2.2) [[X.y].21 + [[y.21.x} + [lady] = 0

As an example, set of smooth vector fields on a manifold is (infinite dimensional) Lie

algebra with the bracket operation defined in Section 1.

If a e G , then the left translation by a and the right translation by a are
respectively the diffeomorphisms La and Ra of G defined by La(x) = a x , Ra(x) = xa .
A vector field X on G is called left invariant if for each a e G, (La )*0 X = X 0 La .
The set of left invariant vector fields on a Lie group G forms a Lie algebra called the Lie
algebra of G and is denoted by g . If we define a map at : g —-) TeG by or(X) = X(e) ,
then or is vector space isomorphism, so dim g = dim G. We can define [ , ] on TCG by
requiring that or becomes Lie algebra isomorphism , thus identifying the tangent space at
the identity of G with the Lie algebra of G.

14

A subspace h of g which is closed under [ , ] is called a subalgebra of g . If h is
a subalgebra of g , then It defines an involutive distribution and the maximal connected
integral manifold H through c is a subgroup of G (which, in general, is not a closed subset
of G ). Conversely, if H c. G is a Lie subgroup, then the tangent space h of H at e is a
subalgebra of g .

If we take 11 to be any l-dimensional subspace of g , then [h, h ] = O c: h . The
subgroup corresponding to such an In is called a l-parameter subgroup. For any X e TeG
we have a natural homomorphism of Lie algebras do : R —-) g with d¢(d/dt) = X , and
hence a Lie group homomorphism (l) : R —-) G mapping R onto the integral curve through
the origin of the left invariant vector field determined by X . We denote ¢(1) by expeX and
this coincides with usual exp defined before for smooth manifolds. The structures of g and
G are related by the exponential mapping, in fact, the Lie algebra determines the Lie group
in the sense that if G and G' are two simply connected Lie groups which have isomorphic
Lie algebras then G and G' are isomorphic.

A Lie group G acts on itself on the left by inner automorphisms Os : G -) G ,

g e G , defined by 08(x) = g x g'1 . The identity e is a fixed point of any such action.
The map
g -> dcgl TeG 53

is a representation (i.e. homomorphism) of G into Aut(g) = GL(g) s GL(n). It is called the
adjoint representation and is denoted by Ad : G —9 Aut(g ). So Ad(g) = ng o 8.1 .
Define ad : g -) gl(g) to be the differential of the adjoint representation, ad = d(Ad) .
Then adX(Y) = [X, Y] for every X, Y e g , and by Jacobi identity (1.2.2) , adX is a
derivation of the Lie algebra g , i.e. adX([ Y, 2]) = [ adX(Y), Z] + [ Y, adX(Z) ] .
Let K be a closed subgroup of a Lie group G, and let G/K denotes the space of
cosets [gK I g e G}. Let 1: :G —-) G/K denotes the natural projection 1t(g) = gK. Then

G/K has a unique manifold structure such that rt : G -) G/K is smooth fibration, i.e. 1t is

 

15

smooth and there exist local smooth sections of G/K in G . We call G/K a homogeneous
space . G acts naturally on G/K on the left by g'rt(g) == x(g'g ) and this action is clearly
transitive hence the name homogeneous space .

Leta:GxM —9MbeasmoothactionofaLiegroquonMontheleft and
denote or(g, p) = org(p). The action is called transitive if for any pair x, y e M there exists
g e G such that org(x) = y . G acts efi’ectively on M if org(p) = p for every p e M
impliesg=e. Let 06 M andlet K= {ge Gla8(o) =o ].KisaclosedsubgroupofG
called the isotropy group at 0. We now state the following theorem (see [W a] ) .

Theorem 1.2.1. Let a : G x M -) M be a transitive action of a Lie group G on a
manifold M on the left . Let 0 e M , and let K be the isotropy group at o . Define a
mapping [3 : G/K -) M by B(gK) = (18(0) . Then [3 is a diffeomorphism .

For each k e K (= isotropy group at o) the map p : K —-> GL(TOM) defined by
p(k) = dockl TOM is a representation of K ( [Wa], p.113) called the linear isotropy
representation and the group p(K) of linear transformations of TM is called the linear
isotropy group at o . Because of the Theorem 1.2.1 we adopt the following definition.

Definition 1.2.1. A Riemannian manifold (M, g) is called (Riemannian)
homogeneous space if the group of isometries I(M) acts transitively on M .

Since there may be more than one Lie group acting transitively on a given
homogeneous space we use the term G - homogeneous if G is a closed subgroup of KM)
which acts transitively on M. Note that M is compact if and only if G is compact. Since an
isometry f is determined by giving only the image f(o) of a point 0 and the corresponding
tangent map df I o , the linear isotropy representation of a Riemannian homogeneous space

is faithqu (injective) orthogonal representation .

We recall that the projective spaces are homogeneous manifolds

mm = SO(n+1)/O(n) , op“: s11(n+1)/S(U<1)U(n)) .

 

16
or": Sp(n+1)/Sp(n)Sp(1) . CayP2= F4/Spin(9) .

A homogeneous manifold M = G/K is called reductive if there is an Ad(K) -
invariant subspace m of g that is complementary to k , g = k O m , where g and k are
the Lie algebras of G and K respectively. All homogeneous Riemannian manifolds are
reductive ( see e.g. [T-V], pp 19-20 ). For the Levi - Civita connection and the curvature
of a reductive homogeneous space see [K-N] and [Be] . Given a homogeneous space G/K
we can define symmetric Ad(G) - invariant bilinear form B : g x g —) R by
B(X, Y) = tr ( adX o adY) . B is called the Killing - Carton form of g . For a reductive
homogeneous space G/K , B is negative definite on k but 111 is not necessarily B
orthogonal to I: nor is B definite on m in general. We state the following theorem which
can be found in [Ch], p. 48 or [O'N] , p. 311 .

Theorem 1.2.2. Let M = G/K be a reductive homogeneous space with Ad(K) -
invariant splitting g = k 6 m . Then the linear isotropy group {dork I k e K } acting on
TOM corresponds under (11: to Ad(K) on m ( 1: is a natural projection G -) G/K) .

Next, we give basic facts about symmetric spaces. For thorough study see [He] .

A Riemannian manifold M is called a symmetric space if for every point p e M ,
there exists an involutive isometry sp with p as an isolated fixed point. Isometry sp is in
fact geodesic symmetry at p, Sp(‘Y(t» = 'K-t) , for every geodesic 7 through p = 7(0) . Every

symmetric space M is a homogeneous space M = G/K , where G = 10(M) is identity
component of isometry group of M and K is a compact subgroup of G ([He], p. 208 ).

For a symmetric space M = G/K , K isotropy group at o , we define involutive
automorphism o : G -) G by o(g) = sogso . Then G; g K (3 Ga where
Ga = {g e G I o(g) =-- g ) and (3:, is its identity component. Automorphism o induces
involutive automorphism of g (by dolToM) denoted by the same letter 0. We denote by It

and m respectively +1 and .1 eigenspace of o . Then I: is the Lie algebra of K , m can be
identified with TM, and we have the following direct sum decomposition

17

(1.2.3) g=k$m , with
[k,k]clt , [m,m]ck , [Ir,m]cm .

Decomposition (1.2.3) is called the Carton decomposition of g with respect to o.
Let(g,k,o)beatriple such that : (1) g isaLie algebra (over R) ; (2) oisan

involutive automorphism of G ; (3) k = F(o, g), the fixed point set of o, is compact

subalgebra . Then ( g , k , o) is called an orthogonal symmetric Lie algebra (o.s.L.a.).

Obviously, for every symmetric space G/K we have an o.s.L.a. associated with it.
Let(g,k,o) bean o.s.L.a. withg =1: 6 m ,andleta beamaximal abelian

subspace of m . Then the dimension of a is called the rank of o.s.L.a. ( g , k , 0').
Correspondingly, the rank of a symmetric space is the maximal dimension of a flat, totally
geodesic submanifold (flat torus) of M. Compact rank one symmetric spaces are sphere and

projective spaces. Compact symmetric spaces of rank two are used in construction of
isoparametric spherical hypersurfaces ( see section 4 of this chapter).

Lie algebra g is semisimple if the Killing - Cartan form B is nondegenerate. An
o.s.L.a. ( g , k , o) with g semisimple is said to be of compact type if B is negative

definite. In that case, - B restricted to m defines Ad(G) - invariant inner product .

3. Second standard immersion of a Sphere

For a good exposition on this topic see [C 4] . On Euclidean space Em we have
canonical inner product < , > given by < u, v > = u‘tv , where vectors u, v e E“[1 are
regarded as column matrices and ut is the transpose of u . The sphere of radius r centered

at the origin is defined as Sm'1(r) = { x e Em l < x, x > = r2 ] . Hypersphere of unit

radius centered at the origin will be simply denoted by S“"'1 .

18

Let SM(m) =-- {P e GL(m;R) l Pt: P} be the space of real symmetric m x m
matrices. Since every symmetric matrix P e SM(m) has m(m+l)/2 independent entries,

SM(m) can be regarded as Euclidean space of dimension N = m(m+l )/2 . Moreover, if
we define metric g on SM(m) by

(1.3.1) g(P.Q)=%tr(PQ) . P.Qe SM(m)

then g is identified with the canonical metric on EN . For computational purposes
2 .

(multiplication of matrices), however, we view SM(m) as sitting in E” . Consrder now

the mapping f : Sm'l -) SM(m) defined by f(u) = u-ut where u e 8“"1 c: E” is a column

vector in Em of unit length . Thus, if u =(u1, , um)t we have

11% ulu2 ulum
2
(1.3.2) f(u) = “201 “2 “2"“

2
umul Dmllz 11m

We see that f is an isometric immersion by virtue of f*(X) = 11 XI + X ut . It is in fact
second standard immersion of 8”“1 and since f(-u) = f(u) it gives an embedding of RP'“.
Since tr f(u) = 2n? = <u, u> = 1 and f(u)2 = u(u‘u)ut = u ut then by comparing the
dimension we seei that f(Sm'l) = { A e SM(m) 1A2 = A and tr A = 1 ]. Thus the image
f(sm-l) is a real projective space lying fully in a hyperplane E1 = [A e SM(m) I tr A = 1)
of SM(m) = EN . We call f(Sm'l) a Veronese submanifold. Also we check that

I I __1_ _2_____
g(A'E’A'h—r) - 2U(A-m) — 2m .

where I is m x m identity matrix, so f(Sm'l) lies in a hypersphere SE30) of SM(m)

centered at I/m with radius r = amal- .The mean curvature vector of f : 8“"1 —-9 SM(m)

19
at u e 8"“1 can be computed ([C 4]) as H = mi_1( I - m f(u) ) which is parallel to the
. I '. . .
radius vector f(u) - E . Thus, f(Sm'1)1s minimal submanifold of a hypersphere Slflrfifi) .

Tangent space and normal space of KS“) are given respectively by

(1.3.3) noosel = ( P e SM(m) l P f(u) + f(u) P = P} ,
(1.3.4) Tflt)sm'1 = { P e SM(m) 1 P f(u) = f(u) P} ,

or, equivalently,

(1.3.5) rfltpm" = ( P e SM(m) 1 Pu = nu , for some it e R } .
If 6 is the second fundamental form of f, then ( see [C 4], [R] )

(1.3.6) 6 (x, Y) = x Y‘ + Y xl — 2 <x, Y> f(u) , x, Y e Tasm-l .

It is known that Bis parallel , i.e. V '6 = 0 .
From (1.3.4) we see that both I and f(u) are normal to 8"“1 via f , also, for any

tangent vector X to sphere, X Xt is normal to 8“"1 . We prove the following lemma that

will be used in Chapter 3 .

Lemma 1.3.1. For a standard hypersphere u : Sm”l -) Em , let f be the second

standard immersion f : 8"“1 —) SM(m) by f(u) = u-ut . If e1, , em_l is alocal
m-l

orthonormal frame of tangent vetors to 8""1 then I = u ut + 2 eieit , where I is m x m

1:]

identity matrix.

Proof. Consider the following matrices : u ut , eke]: (1 S k S m-l), 6,6; + 61-6:

(1 S i < j S m—l) . By (1.3.4) they all belong to the normal space TtSm'1 , and there are

20

 

-1 2- +2 .
1 + m - 1 + (m2 ) = m I; of them 1n number. Also these vectors are linearly
independent ( they are mutually orthogonal ) . On the other hand, dim SM(m) = m

0 - o 2-
and drmTuSml =m-l, so,dtrnrtsm-1= “”3“ -m+l= m 3‘” .We

conclude therefore, that 'I‘;';S""'l = Span{u u‘, eke: , eiejt + ejei‘} . In particular,

 

I = a (u u‘) + Z bk(ekei) + Z cij(eiejt + ejeit) . Using (1.3.1) , it is easy to see that
1 K]

cij = 0 and a = bk = l for every k , proving the lemma. 0

Standard embeddings of projective spaces can be realized in an analogous way
using Hopf fibration . Namely, let F denote one of the fields R of real numbers, C of
complex numbers or skew field Q of quatemions, and let d = d(F) be the dimension of F
over the reals. For a matrix A over F , At and A denote transpose and conjugate matrix
and let A“ = At . M(m; F) is the set of all m x m matrices over F and the set of Hermitian
matrices is H(m; F) = { A e M(m; F) IA* = A}. F‘" is considered as an md - dimensional
vector space over R with the usual Euclidean inner product < z, w > = Re(z*w) . All
vectors in Fm are regarded as column matrices .

Projective space FP’"‘l is considered as the quotient of the unit hypersphere
sm‘“= { ze F‘“lz*z= 1) obtainedbyidentifying zwith zxwhcre he F with 12.1 = 1.
FP""l is given canonical metric such that 11: : Smd‘l —) FP""l is a Riemannian submersion
with totally geodesic fibers . Note that we have natural action of the unitary group U(m; F)
on FP""1 induced from the one on the sphere Sm‘l'l . Define the map

q):1=P""l —) H(m; F) by

¢(p) = 22* , where 25 1t'1(p) .

This map is well defined and gives an embedding of FP""1 into H(m; F) (the first
standard embedding of a projective space). The image of FP""1 under this map is given

21

as ¢(FP“"1)={Ae H(m;F)|A2=A and trA=1} andliesasaminimalsubmanifoldin
a hypersphere of H(m; F) centered at Ill!) and with radius r = 31—1 . The Cayley

m

projective plane CayP2 cannot be realized via Hopf fibration and is simply defined as
CayP2 = {A e H(m; F) 14.2 = A and trA = 1} . Embedding b was first studied by Tai
[Tai] , who proved that the embedding q) is equivariant with respect to and invariant under
the action of U(m; F) . For other properties of this map see also [S], [R], [C 3] and [C 4].

4. Isoparametric spherical hypersurfaces

In this section exposition follows essentially [Ce-Ry], [Car 2-5], [M], [F], [N 1-2]

and also uses results of [T-Ta], [T 3], [H], [H-L], [A] .
Originally, a family of hypersurfaces M? in a real space form M“+l(c) of constant

sectional curvature c is called isoparametric if each M}1 is equal to level hypersurface

f1(t) where f is a non - constant real valued function on M"+I(c) which satisfies system of

differential equations of the form

llVf 112 =a(f) , if = b(f)

for some smooth real - valued functions a, b . Thus, the two classical Beltrami differential

parameters, square of the norm of gradient and Laplacian, are functions of f itself, whence
the name isoparametric. (For the shape operator and mean curvature of such level
hypersurface in terms of a and b see [Ce-Ry] or [Ha] ). Equivalently, an isoparametric
family of hypersurfaces can be characterized as a family of parallel hypersurfaces, each of
which has constant principal curvatures ( [Car 2], [N 2] ). We will adopt the following

definition .

22

Definition 1.4.1 A (complete) hypersurface is called isoparametric if its principal
curvatures (and their respective multiplicities) are constant.

Canan [Car 2] established the following basic identity for principal curvatures of an
isoparametric hypersurface of a space form M“+I(c) .

Theorem 1.4.1 Suppose that an isoparametric hypersurface M has v distinct

principal curvatures k1, , kV with respective multiplicities m1, , mV . Then

(1.4.1) inc—3151-Iii =0 . lSiSV.
Jki-kj

j¢i

Using this key identity Cartan was able to determine all isoparametric hypersurfaces
in the cases c s 0 . Actually if c S 0 , then there are at most two distinct principal
curvatures of M and M is either umbilical (one curvature), or standard spherical cylinder
Sk x En'k (standard product Sk x Hn'k) in En+1 for c = 0 (respectively in hyperbolic
space form Hn+1 for c = -1 ).

For hypersurfaces of the sphere 8'”1 things are much more interesting, in
particular number of principal curvatures can be greater then two. E. Cartan undertook
study of the spherical isoparametric hypersurfaces in the series of papers [Car 2 - 5] . He
classified isoparametric hypersurfaces of Sn+1 with two distinct principal curvatures as
standard products of two spheres [Car 2] , and he found that those with three distinct
principal curvatures are precisely the tubes of constant radius over the standard embeddings
of PP2 for F = R, c, Q(quaternions), O(Cay1ey octaves) in 84, 87,513, 525 respectively
[Car 3] . In each isoparametric family of parallel hypersurfaces there is a unique
hypersurface which is minimal in sphere. It is easy to see that the principal curvatures of

minimal isoparametric hypersm'faces with three principal curvatures are ‘1—3 , 0 , - ~13 , i.e.

23

they are roots of the equation x3 - 3x = o . Namely, from minimality and (1.4.1) with

c =1 we have

3ki-lt?-2detA = o , i=1,2,3

from which ki 6 N3 , 0 , - «f3 }. We also used the fact that the multiplicities of principal
curvatures for isoparametric spherical hypersurface with three curvatures are the same: 1,
2, 4 or 8 in dimensions 3, 6, 12, 24 respectively [Car 3] . Isoparametric spherical
hypersurfaces with three principal curvatures are all homogeneous. They are identified as
30(3)/z2 x 22 , SU(3)/r2 , sp(3)/sp(l)3 , F4/Spin(8) of dimensions 3, 6, 12, 24
respectively (see [H-L] , [T-Ta] ). The minimal hypersurface of the type SO(3)/Zz x 2,2 in

S4 we call the Cartan hypersurface .

Cartan showed that any isoparametric family with v distinct principal curvatures of

the same multiplicity can be defined by the equation

F = cos v t (restricted to Sn+1 )

where F is a harmonic homogeneous polynomial of degree v on En+2 satisfying
llgrad F||2 =-. v2 r2v-2 ,

where r is the distance from the origin and gradient is in E“2 . For example, for

hypersurfaces with 3 principal curvatures polynomial F is given by (cf. [Car 3])

(1.4.2) F=u3— 3uv2 + %u(XX+YY—ZZE)
+ 2%? v()5(—YY)+ ¥3 (XYZ+ iii).
In this formula 11 and v are real parameters, while X, Y, Z are coordinates in the algebra
F = R, C, Q, 0 respectively for the cases corresponding to the multiplicities m = 1, 2, 4, 8

24

The sum XYZ + EYX is twice the real part of the product. In the case m = 8 ,
multiplication is not associative but the real part of XYZ is the same whether one interprets
the productas (XY)Z or X(YZ).

In [Car 5], Cartan gave examples of two families of isoparametric hypersurfaces in
85 and 89 with four distinct principal curvatures of the same multiplicity (respectively 1

and 2 ). The one in S5 has particularly nice representation by the map
31 x 33,2 -+ s5 c E6 given by

(1.4.3) (0 , (x,y)) -> ei9(cost x +isint y)

Here, 83.2 denotes Stiefel manifold of orthonormal pairs of vectors in E3 and S1 is the
unit circle . More precisely each isoparametric hypersurface M? c 85 with four principal
curvatures is the irnmage of the map (1.4.3) which doubly covers M? . The minimal one is
obtained when t = lt/8 [N 2] . Nomizu used this map to construct infinite family of
isoparametric hypersru'faces M2,n with four principal curvatures of multiplicities 1, n—l ,
1 and n-1 . Takagi has shown ([T 3]) that any isoparametric hypersurface with four
curvatures such that the multiplicity of one curvature is 1 is congruent to the example M2,n
of Nomizu for some n and t .

All examples of isoparametric spherical hypersurfaces known by Cartan are
homogeneous. In fact each is the orbit of a point under an appropriate closed subgroup of
SO(n+2). Of course such orbit hypersurfaces have constant principal curvatures [T - Ta] .
In particular, isoparametric hypersurfaces with four principal curvatures of the same
multiplicity 1 or 2 mentioned above are SO(2) x 80(3)]sz , respectively Sp(2)/l‘2. The
minimal hypersurfaces in these two families have principal curvatures equal to ‘12 + 1 ,
‘12-1,1-~I'2,-\f'2-1(rootsof x4-6x2-1-1 = 0)andtheycanbefoundinasimilar

way as was done in the case of hypersurface with three curvatures, using identity (1.4.1).

25

Cartan did not know what the possibilities were for the number v of distinct
principal curvatures, nor whether isoparametric hypersurface is necessarily homogeneous.
Work on isoparametric spherical hypersurfaces was revived by Nomizu [N 1-2] and then
several important results followed. Using classification of [H-L] Takagi and Takahashi
determined all homogeneous hypersurfaces in sphere (including some with 6 curvatures)
and found their principal curvatures [T-Ta]. Ozeki and Takeuchi ([O-Tl) Produced two
infinite series of isoparametric hypersurfaces which are not homogeneous. Major results in
the theory were obtained by H. F. Mi'nzner . Through a geometric study of the focal
submanifolds of an isoparametric family and their second fundamental form he reproved
Cartan's identity (1.4.1) showing it to be equivalent to the minimality of focal
submanifolds (Left hand side of (1.4.1) is trace of the shape operator of a focal

submanifold). He also proved the following theorem [M] .

Theorem 1.4.2. If kl > k2 > > k, are distinct principal curvatures of an

isoparametric spherical hypersurface with respective multiplicities m1, m2, , m\, then

ki=cotei , 0<91<...<9v<1t

whcrc ei=91+flfi, ISISV , Wlth 91<£ ,
V V

and the multiplicities satisfy mi = mm (subscripts mod v ) .

As a consequence, there are at most two different multiplicities m1, m2 for principal
curvatures and if v is odd then all multiplicities must be equal. (Miinzner was also able to
show that if v = 6 then m, = m2 ) . Using delicate cohomological arguments he also proved

the following splendid result .

Theorem 1.4.3. The number v of distinct principal curvatures of an isoparametric

hypersurface satisfies v = 1, 2, 3, 4 or 6.

26

Generalizing Cartan's result, Miinzner showed that the hypersurfaces of any
isoparametric family with v distinct principal curvatures in SMl can be represented as open
subsets of level hypersurfaces in Sn+1 of a homogeneous polynomial F of degree v on

En+2 which satisfies the differential equations (on 5“”)

llgrad P112 = v2 xiv-2
(1.4.4)

= V2(ml’ ml) I.v-2

AF 2

 

As a consequence, every isoparametric hypersurface is algebraic, and a piece of
isoparametric hypersurface can always be extended to a complete one . Let us state also the

following result of Abresch [A] who used refined techniques of Mi'mzner to prove

Theorem 1.4.4. i) Given an isoparametric hypersurface in 8"“1 with v = 4
principal curvatures , let m1 5 m2 be (possibly same) multiplicities of curvatures . Then
the pair ( m1 , my) satisfies one of the three conditions below

(a) m, +m2+ 1 isdivisibleby 2s :=min [2°|2°> m1,o e N}.

(b) m1 is power of 2, and 2m1 divides m2 + 1 .

(c) m1 is power of 2, and 3m = 2( m2 + l) .

Each condition corresponds to a topologically different kind of examples .

ii) Given an isoparametric hypersurface in 8'”1 with v = 6 then m, = m2 6 {1, 2} .

Regarding isoparametric hypersurfaces with four curvatures of the same
multiplicity, Cartan asserts , without proof, that they have to be homogeneous [Car 5].
That was proved by Ozeki and Takeuchi if m1 = m2 = 2 [OT]. However, in the light of
the above theorem of Abresch we can easily prove that statement and moreover completely

classify isoparametric hypersurface with four curvatures of the same multiplicity. Namely,

27

if m, = m2 then case (b) of the theorem gives m1 = m2 = l and then the results of Takagi

[T 3] and Takagi and Takahashi [T-Ta] classify such hypersurface as SO(2) x SO(3)/Z2 .

If case (c) occurs, then m = m2 = 2 hence by the result of Ozeki and Takeuchi [04‘] the

hypersurface is homogeneous and therefore according to the list in [T-Ta] must be
sp(2)/r2 . Therefore these hypersurfaces are exactly those two found by Cartan in [Car 5] .
Next, we give the list of all isoparametric hypersurfaces in sphere with three or four
distinct principal curvatures of the same multiplicity. As remarked by Hsiang and Lawson
[H-S] , homogeneous isoparametric hypersurfaces in sphere arise from isotropy
representations of the corresponding symmetric spaces of rank 2. For our hypersurfaces,
their isometry groups G, actions 11! , principal isotropy groups H, common multiplicity of
principal curvatures m and dimension n are given as follows (first four examples in the

table have three curvatures, remaining two have four ).

Table l. IsOparametric hypersurfaces in sphere with three or
four principal curvatures of the same multiplicity

-_—fl-

 

 

 

 

 

30(3) 529;, - e g x 22L 1 3
SU(3) AdsmL T2 2 6
Sp(3) A2V3 - e Sp(l)3 4 12
F4 9L Spin(8) 8 24
80(2) x 30(31 piglp3 21 1 4
3132) Ad T2 2 8

 

 

 

 

 

 

 

Let us mention at the end that the theory of isoparametric hypersurfaces continues to

be area of active research. Subsequent investigation exploited equations (1.4.4 ) of

28

Miinzner - Cartan and new results were obtained using algebraic tools such as triple
systems , Jordan algebras, Clifford systems (cf. [D-N], [F-K-M], [W 2] ) . For example,
Ferns, Karcher and Mi'nzner gave a construction of isoparametric hypersurfaces with v = 4
using representations of Clifford algebras which included all known examples, except two.
Their method also exibited infinitely many series of infinite isoparametric families with four
constant principal curvatures. However, the main problem of classification of isoparametric
hypersurfaces in sphere still remains open. For isoparametric hypersurfaces in pseudo -
Riemannian space forms see [Ha], [N 3] and [Ma] , and for real hypersurfaces with
constant principal curvature in complex projective or complex hyperbolic spaces see [W 1],

[T 2], [B] . One possible generalization to a submanifolds of higher codimension was dealt

with in [T e] . See also [Pa-T] .

5. Finite type maps and submanifolds

For spectral geometry standard references are [B-G-M] , [Ch 2] and for finite type

submanifolds [C 4] .
Let (M", g) be a Riemannian manifold. Laplacian A acting on smooth functions is

defined as

(1.5.1) Af = 2[(Vciei)f—ei(eif)] , fe C°°(M)
i=1

where lei] denotes local orthonormal basis of tangent vectors (A does not depend on the

choice of such basis). In local coordinates, A has the following expression

(1.5.2) At = - —1- Xaj(gik\1§a,i) , where g =det(gij) .
lg— 12k

29

The following property of A acting on the product of two functions is well known

(1.5.3) A(uv) = (Au)v + u(Av) - 2<Vu,Vv> , u,ve C°°(M) .

The Laplacian is naturally extended to act on Em - valued maps (componentwise), so the

rule above extends to inner product of vector functions U, V on M as follows

(1.5.4) A< U,V>.=<AU,V>+ <U,AV> -2;<Vc,UaVe,V>

(1.5.5) A(fU) = (ADU + f(AU) - 22(eioi'76iu , re C°°(M)

Also, if x : Mn —-) E"[1 is an isometric immersion whose mean curvature vector is H, then

the following formula holds (see e.g. [C 4], p.135)

(1.5.6) Ax = —nH

An eigenvalue of A is any real number 3. for which there exists a smooth nonzero funcion
f (called an eigenfunction ), so that Af = If . The set of all eigenfunctions of A , V1 ,
forms a vector space and its dimension (need not be finite) is called multiplicity of 2. .
Clearly, for two different eigenvalues hp , 1‘! we have Vpn Vq = {0] . The set of all
eigenvalues taken with their multiplicities is called spectrum of M and denoted by Spec(M).

If M is compact, we can define natural L2 - inner product ( , ) by (f,g) = f f ng .
In this case Laplacian is self adjoint strongly elliptic operator, all eigenvitalues are
nonnegative and the spectrum is discrete , Spec(M) = { 0 = ko< kl< 1.2 < T on }.
Multiplicity of each X (dim V1) is finite, dim V0 = 1, and 2V, is dense in C°°(M) .

t.

Thus, we can write

(1.5.7) C°°(M) = iv, (in L2 - sense) .
t=0

30

This direct sum decomposition is orthogonal with respect to ( , ) . According to (1.5.7) ,
any smooth function f e C°° (M) has the following specu'al decomposition

(1.5.8) r = to + 2i, (in 1}— sense) ,

where f0 is constant ( Afo = 0 ) and fl is the projection of f onto Vt , i.e. Aft = 3., ft .

A map f e C°°(M) is said to be finite type map if its spectral decomposition
(1.5.8) has finitely many nonzero terms. More precisely, f is of k - type if there are
exactly k nonzero terms f 1’ , fq‘ ( tiz l, i = 1, ..., k ) in the decomposition (1.5.8).
The set {t}, , tk] ( also [7111, , 21k] ) is called the order of a map f . If f is not of
finite type, that is, decomposition (1.5.8) has infinitely many nonzero terms, f is of
infinite type .

Note that A can be naturally extended to E"1 — valued maps (by taking Laplacian

componentwise), and accordingly, we extend the notion of finite type map as follows. For
a smooth map x : Mn —) Em , x = (f , , fm) , we find spectral decomposition (1.5.8)

of each fi and combine them to obtain spectral decomposition of a vector function x as

(1.5.9) x = x0 + Zn, (in L2- sense) ,

where, xI = ((fl)t , , (fm)t) , i.e. Axt= it, xt . (Some of the (fi)t's can be 0). Again ,
vector function x is called k - type if there are k nonzero vectors x, (t 2 1) in decomposition

(1.5.9). In particular, a submanifold of Em is of finite type (k - type) if the corresponding

immersion is so. x0 is always a constant vector, and if x is an isometric immersion of a

compact manifold M, then 110 is the center of mass of M in Em , i.e. x0 = Eli—MI jx .
M

31

If x : M’1 —-) S?‘1(r) c: E‘m is an isometric immersion of a compact manifold into a
sphere, then Mn is called mass - symmetric in Sf'l(r) if x0 = c , i.e. center of mass of M
coincides with the center of sphere. For an 1 - type immersion x : M" —) Em , we have

x = xo+ xp with x0= const , A xp = AP xp . The well known theorem of Takahashi

[Ta 1] canbestatedintermsofl -typemapsasfollows
Theorem 1.5.1. Let M be a compact submanifold of E‘“ . Then M is of 1 - type
if and only if M is a minimal submanifold of a hypersphere of Eml .

If x1 : Mn -—) Em1 and x2: M“ —-) Em2 are two isometric immersions, then the
.1.
a]?
are of finite type. Let M be a compact, irreducible symmetric space and p1 < p2 < < pk

diagonal immersion x = D(xl, x2) = (x1, x2) is of finite type if and only if both x1, x2
any finite set of natural numbers. Then the diagonal immersion D(xpl, , ’59 of the
standard immersions xpl, , xpk is of k - type with order [p,, p2, ...,pk] . This shows
that there are immersions of arbitrary high type. Also, if M is a compact homogeneous
space which is equivariantly, isometrically immersed in E” , then M is of k - type with
k S m ([C 4], p. 258 ; see also [Ta 2] and [D] ) . A closed curve C in Em is of finite type if
and only if Fourier series expansion of each coordinate function of C has only finitely
many nonzero terms ([C 4], p.283 ). We give the following criterion for finite type

immersions [C 4] .

Theorem 1.5.2. Let x : M —-) Em be an isometric immersion of a compact
Riemannian manifold M into Em . Then M is of finite type if and only if there is a non -

trivial polinomial P(t) such that

(1.5.10) P(A) (x - x0) = 0 .

Moreover, M is of k - type if and only if polinomial P is of degree k having exactly k
distinct (positive) roots and for any other polinomial Q that satisfies Q(A) (x - x0) = 0 , P

32

is a factor of Q . The statement of the theorem remains the same if x - x0 is replaced by the

mean curvature vector H .

Let us note that the notion of finite type map and irm'nersion make perfect sense also

for a noncompact manifold, e.g. an immersion x : M —) E“1 is of k - type if we can write
(1.5.11) x = x0+xtl+m+xtk ,

where x() is a constant vector and xtl, , xtk are eigenvectors of the Laplacian
corresponding to k different eigenvalues let], , 711k . If M is noncompact , X's need not
be positive, nor their multiplicities finite. Eigenspace V0 (set of harmonic functions) is
generaly of dimension > 1 (there may be nonconstant harmonic functions). If one of the
eigenvalues 1,1, , 31“ corresponding to the decomposition (1.5.11) is 0, then the
submanifold is said to be of null k - type. In this case x0 is not uniquely determined ( for
compact manifold, x0 is always center of mass). The cylinder x(O, u) = (c089, sine, u )
is an example of noncompact null 2 - type submanifold.

Notions of order of a submanifold and submanifolds of finite type were first
introduced by B.Y. Chen in [C 2] and the theory of finite type submanifolds has become an
area of active reseach (see [C 4]). In particular, there is a problem of classification of low
type submanifolds which lie in a hypersphere. By Theorem 1.5.1 , l - type submanifolds
are characterized as being minimal in sphere and one can expect that 2 - type and higher
type submanifolds are more general. Indeed classification of even 2 - type spherical
submanifolds seems to be virtualy impossible.( Note, however, that the only compact
2 — type surface in s3 is flat torus 81(a) x s‘(b), a it b [B-C-G] ). On the other hand,
studying finite type immersions of a spherical manifold into SM(m) via the second standard

immersion of the sphere proved to be more manageable ( see [R], [B-C] ) . In Chapter 3

33

we study spherical hypersurfaces which are of 2 - type and those which are of 3 - type and

mass - symmetric via the second standard immersion of the sphere.

CHAPTER 2

QUADRIC REPRESENTATION OF A SUBMANIFOLD

For an isometric immersion x : M“ -) E’“ of a Riemannian manifold into a
Euclidean space, one defines the map 32' : M“ -) SM(m) from M into the set of real
symmetric m x m matrices by 'x’ = x-xt , where x is regarded as a column vector in E‘“ .

Thus, if x = (x1, , xm)t we have

xf xlxz xlxm
xmxl xmx2 xx":l

We call f the quadric representation of a submanifold M. In this chapter we establish
some general results about the quadric representation. First we prove a theorem about

quadric representation being an isometric immersion.

Theorem 2.1. Let x : M“ —) Em be an isometric immersion of a Riemannian

t

manifold into a Euclidean space. Then 32' = xx is an isometric immersion if and only if

x(M“) c 8““1 , i.e. M is spherical. (In the case n = 1, a curve is assumed to be complete.)

Proof. First we prove the statement for a complete curve x : C —) E’“ . Let

x(8) = (x1(s), x2(s), , xm(s)) be the parametrization of the curve by its arclength . Then

34

35

d§®d§=g(d§,d§)= u(d'x'd'x') = %2(xixj+xixjf)2ds2
r,j

NIH

Slnce x is assumed to be an isometry, tangent vector d'r'E/ds must have length 1,

therefore we get

1 r I
1 = 2 §0qu + xixj)2
1 v ,
= 2(2’5’92 + 2 20% ' "i"?2
l l,j

= 2(2’93‘92 + 2(Xi’X - X9532

i<j

= %[(uxuz)']2+ ll “it'll2
= %[(uxuz)']2 + l|x||2 - %[(llx||2)’]2

= llxll2 + -‘li-[(llxll2)’]2.

Here, A represents the usual wedge operation in the Grassmann algebra over Em. Hence

for v1 ,v2 6 B” we have
2 2 2 2
lllevzll = det(<vi,vj>) = llvlll llv2|l — <vl , v2) .

Thus letting u(s) = llxll2 we get the differential equation in u that separates the variables,
11 + (1/4)(u’)2 = 1 . One obvious solution is u = 1 , and there is no solution for u > 1 . If
u it 1, solving the equation gives u(s) = 1 - (c + s)2 , where c is an arbitrary constant. This
solution, however, represents decreasing function of s and therefore, u = llxll2 < 0 for
sufficiently large s (curve is assumed to be complete) which is a contradiction. Therefore,

u = 1 , i.e. curve C belongs to the unit sphere centered at the origin.

 

36

Now let 'x' be an isometric immersion for a manifold M“ (n > 1). Since it preserves
the first fundamental form of M, it also preserves the first fundamental form of any curve
of M (isometry property is hereditary to a submanifold). Let p e M be an arbitrary point,
and consider a small smooth loop based at p. Such loop can be chosen as the image of a

circle passing through p in the normal neighborhood in the tangent space TpM via the
exponential map. The restriction of 'x' to this loop is an isometry, and from the above we
conclude that the loop belongs to the unit sphere centered at the origin and the same is true

for point p. Since p is an arbitrary point of M , M is a spherical submanifold. The converse

of the statement is well known. 6

Actually, we have a similar result under weaker assumptions .

Theorem 2.2. Let x : M“ —) E‘“ (n > 1) be an isometric immersion. Then 'i' is
a conformal map if and only if M“ c S’“'1(r), in which case i' is homothety.
Proof. Let 'g and < , > be metrics on SM(m) and M respectively, and V and V

beEuclidean connections on SM(m) and E‘“. Ifwe set 32' = (fl, , fN) , where

N = dim SM(m) , then

(if (X) = (df , ,di)x = (df1(X),...,di(X)) = (Xf1,...,XfN) = 6X32 .

Since V acts as a derivation on the set of smooth functions on M then the product rule

extends also to the map 52' = x-xt . namely, we have
9,5 = $54an = (Vxx)) x‘ + x (Vxx)t = x x‘ + x x‘.

If 'x' is a conformal map then i'*g = q) < , > for some positive function 4) . In particular,
i maps a pair of perpendicular vectors into a pair of perpendicular vectors. Therefore, if

X .L Y is a pair of perpendicular vectors of M , we have

37

o = g(ft',x,i,,Y)
= g(di (X).di (Y))
= g(in', 7Y3?)
(2.1) = g(xit‘+x X‘,Yx‘+xY‘)

%U(th+xXt)(Yxt+th)

<x,X><x,Y>+<x,x><X,Y>

<x,X><x,Y>.

If X, Y is a pair of perpendicular unit vectors then X + Y and X - Y are also perpendicular,

and from the equation above we obtain
0 = <x,X+Y><x,X-Y> = <x,X>2— <x,Y>2.

Then (2.1) implies <x , X> = 0 , for every tangent vector X of M, and therefore

X<x , x> = 2<x , X) = 0 , that is , <x , x) = r2 = const, which shows that
x(M) c S'“'1(r) . Converse is easy, because then g( 32*X , i*Y) = <x, x> <X, Y> .0

Now we want to examine some relationships between the map 3? and the condition

of being of finite type. First, let us fix the notation. Let M“ be a submanifold of the

Euclidean space E'“. Suppose that e1, e2, , en, en +1, , cm are local orthonormal

vector fields along M such that the fu'st n vectors are tangent to M and the remaining m-n
vectors normal to M. Let g and V be the Euclidean metric and connection of E’“, and
denote by V , h , D , A: respectively, the induced connection , second fundamental form

of M, connection in the normal bundle T‘LM and the Weingarten endomorphism relative to

38

the normal direction 5,. The connection forms a): and the mean curvature vector H of M in
E‘“ are defined by Vekei = ;m§(ek)e. , H =(1/n)z,‘,(trAr)er . Here, indices i, j, k range
from 1 to n and indices r,s range from n+1 to m . As usual, A denotes Laplacian on M .
The metric on SM(m) is given by g(P, Q) = % tr (PQ) .

Since an l-type map is next simplest to being harmonic , we start out by proving a

theorem about i' being an l-type map .

Theorem 2.3. For an isometric immersion x : M“ —-) E’“ , 32' is of l-type if
and only if M“ is totally geodesic submanifold of the hypersphere S'“‘1(r) c E‘“ . In
particular, if the immersion x is full and M complete, then M = Sm'l(r) is the standard

sphere.

Proof. Suppose that 'x' is l-type map . Then we can write 52' = 360 + Sip , where

550 is a constant vector and Aip = hp '56,, , i.e. it", is an eigenvector of the Laplacian. Thus ,
(2.2) A3? = xprp=xp(t -'x'o).

On the other hand,

A}? = A(x'xt) (Ax) xt+x (Ar)t — zzfieixxiieix)‘
(2-3) =—n(th+xH‘)—22l;ele: .

Therefore, from (2.2) and (2.3) we have

-n(Hx‘+xH‘) — 229%} = 1,,(75 - 3:0).

39

Differentiating this relation along an arbitrary vector field X of M, we get

n[(AHX) x‘ + x (AHX)‘1 — [(DXH) x‘ + x (DxHh
- n (H Xt + X H‘) - 2 2 (015(X)(eiejt + ejeit)
1.1“
— 22m(X,ei)ef + c,h(X.c,)‘l

= M(th+xXt)

Note that the first sum is equal to 0 , since (03(X) is antisymmetric in i and j whereas

1 t . . . . .
ch + ejei ls symmetric 1n 1, j. Also

ZlhOKeQe,t + e,h(X.c,)‘l = 2g(h(X.ei).e,)(cie§ + 9.9:)
l 1.!
= 2 g(A,X, ei)(eie: + e,eit)
1,!

= Z Mme: + ¢r(A,X)‘] .
and therefore, for every X e l"(TM) we have
n[(AHX) xt + x (AHX)‘] — [(DxI-Dx‘ + x (DXH)‘]

(2.4) - n (H xt + x H‘) - 2 2 [(Apoe: + e,(A,X)‘]

= M(x Xt‘l' X Xt) .

We now find eie: + are} component of (2.4) , i.e. appl)I §( - , eiei + 6,61) to it I

40
n g(AHX. ci) g(X. c,) - n g(DxH. e,) g(X. 9,)
- n g(H: er) g(X, Ci) _' 2g(Arx 9 Ci)

= A, g(X. e.) the e,) .
Letting X = e].l and summing on i we get
n(trAH) g(x, er) - n g(DxTH, e,) - n2g(H, er) — 2 HA, = n 2., g(x, er) .
If we multiply this relation by er and sum on r , we obtain

(2.5) (trAH -).p)rN - DXTH = (n+2) H.

Note that in general x is not perpendicular to M so we have normal and tangential

component of x :

XN = 286.999, . XT = Escher,

Finding ere: component of (2.4) and summing on r we get

(2.6) < DXH, xN> = 0 i.e. DXH .L xN for every x 6 TM.

Finding em}, + esefi component of (2.4) and summing on s (after multiplying by e.) we

obtain

<DXH, et>xN + <x, er>DxH = 0 ,

and by (2.6) we have

41

(2.7) <DXH, er>xN = (X, er>DxH = 0, foreveryrand Xe TM.

Thus, at any given point of M we have
(2.8) xN=0 or xN¢0 and DH=0

Next, by comparing eke}, components of two sides of the equation (2.4),

multiplying by ck and summing on k we get

(2.9) <nAHX Jch, xT> =0 , forevery XeTM ,

and by comparing eiefi + ekeit components, summing on k and taking (2.9) into account

we have

(2.10) [n<AHX,ei>—).p<x,t:i>]xT = (nAHX -APX)<x,ci> = 0 ,

for every i = l, 2, , n and every X 5 TM . Therefore, at any given point of M we

have

(2.11) xT=0 or xTatO and nAH=7LpI.

Let U= { pe MI rel-#0 atp } .ThenUisanopen subsetofM,andonUwe
have by (2.11), tr AH = hp . Then (2.5) implies DxTI-I = - (n + 2)H on U . Now let V

beanopen subset ofUdefined by V= { pe U I xNan atp }.By (2.8) we have

DH = 0 on V, and from the above we conclude H = 0 on V , i.e. V is the piece of M

immersed minimally in E‘“. Now we compute tr (A? ) on V, noting that Laplacian

commutes with trace since it is a linear operator.

'42

u(Ai) = A(tri) = A<x,x>

2<AX,X> - 22(3. 3.)
1

1’1

—2n <H,x> —2n

= - 2n
On the other hand, (2.2) yields
u(Ai‘t') = )tp(tr'x' - tr‘x'o) = xp(<x,x> — tr‘r‘t'o).
Therefore, Ap( < x, x > — tr 530) = — 2n , and since obviously hp at 0 we have

~

(2.12) <x,x> = trxo — Q = const

Consequently, x(V) c Sm‘1(r) and hence 0 = H = H' 7)}; , where H' is the mean
curvature vector of V in Sm'1(r) , which is a contradiction because H' .L x and x ¢ 0 .
Therefore, we must have V = Z , and hence on U x = xT is tangential . Now on U as

before we have ( note H J. x)

u(A‘iE) = —2n<H,x> —2n =—2n = 2p(<x,x>— trio).

and therefore (2.12) holds again on U . So, x(U) c Sm'l(r) but then xT = 0 since x is

normal to U for spherical submanifold and this is a contradiction. We conclude U = Q

and x = xN is normal to submanifold M. Consequently, x immerses M into a

hypersphere of E‘“ centered at the origin , x: M“ —-) S’“‘1(r) c E‘“ . In that case

43

O x 0 l I
H=H-F and DH=DH=DH . From(2.8)weget DH=D'H‘= 0 andthen
from (2.5) it follows (trAH - hp) x = (n + 2) (H‘ - 5‘2) . Since H' .L x we see that

H' = 0 i.e. M is minimal in the hypersphere .With these identities in effect, equation (2.4)
becomes (we take e,m = x/r)

m-l
2 “+1 (x xt + x x‘) - 2 2mm; + e,(A,X)‘] = 1.],(x xt+ x x‘) ,

for every X 6 TM . Therefore, AP: 2(n +1)/r2 and A, = 0 for every s = n+1, , m-l.
We conclude that M“ is totally geodesic in Sm'1(r), i.e. it is (a piece of ) standard S“(r) in
sm'1(r) .

Conversely, if M“ is totally geodesic S“(r) c sm-1(r) than it is well known that M“
is minimally immersed via 3? (after scaling the metric in SM(m) with the factor 1/‘r‘2 ) as a
Veronese submanifold in a hypersphere of SM(m) (see Ch.1, Sect.3). Then the well
known theorem of Takahashi (Theorem 1.5.1) asserts that 'x' is of l-type. As a matter of

fact we have

s: = tot; = n—l-+1(xx+r22eii‘e) + n+1(mot -r22e.e.) .

where (xxt + r22 eieit ) is a constant vector , actually equal to r21n+1 in SM(m) by
1
Lemma 1.3.1 , and (n xxt — r22 eiei‘) is an eigenvector of the Laplacian corresponding
1
to the eigenvalue it}, = 2(n + 1)/r2 . Since this is the second nonzero eigenvalue of the

sphere, it follows that S“(r) is of order [2] . ’

It is known that a closed curve in E‘“ is of finite type if and only if its Fourier series
expansion has finitely many nonzero terms (see c. g. [C 4]).There are nonspherical closed

curves in E’“ of finite (see [C 4], pp 288 — 289 and [C 5], pp 16 - 18). They are also of

44

finite type ( i not an isometric immersion) in SM(m) via 32' since by the product formulas of
trigonomedy their Fourier series expansions still have finitely many nonzero terms. Also,
given any finite type spherical submanifold M which is also of finite type via 32' , translate
M by any vector v , so that v + M again belongs to a sphere (now not centered at the
origin). Quadric representation of such translated manifold will no longer be an isometric

immersion, but it will still be of finite type. We also have the following example

Example 2.1 Given two nonspherical finite type curves C1, C2 mentioned above,

consider their product Cl x C2 . Such product does not belong to any sphere and its

quadric representation is of finite type since the Laplacian of a product splits into the sum of

Laplacians on the component manifolds .

However, we are able to prove the following theorem for minimal submanifolds .

Theorem 2.4. For a minimal immersion x : M“ —> E’“ , quadric representation i

is of infinite type .
Proof. Supose 52' is k-type map where k is finite. Then we can decompose f as

x = x0 + xtl+ xt2 +...+ xtk , where x0=const andAxti =7tLi xti.
Finding successively iterated Laplacians of 'x' we obtain
Ax = 2“le + 7112 x12 +...+ Mk xlk
A x -- At] XI] '1' 112 xtz + ... + Atk xtk

Eliminating in , 2,2, , it], from these k+1 equations we get

45

(2.13) 01(1? -X0) '1' Gk_1A(X -X0) +...+ O'l'Ak-RX -X0) + AR; -X0) = 0

where oi is the ith elementary symmetric function of 1,1 , 7c , 7111‘ that is

t2 , ...

01: -( ltl+...+ Mk)

ok-1— (-1)"'1 Z knuxjultk (A denotes omission)
1'

01‘ = (-1)k 11,1 7a,,

As before we find tr (A3?) to be

u(Ai)= A(tri) =A<x,x>= 2<Ax,x> — 22<ei,ei> = -2n
1

and by iterating we get tr ( Ai 'x' ) = 0 for i2 2 . Hence if we take trace of (2.13) we

obtain
(2.14) ok(tr§ -t1'i'0) -2nok_1 = 0 .

1°. If O’k at 0 i.e. submanifold is not of null k-type then

,., ZnQH ..
<x,x>=trx= + tr'x0 =const ,

so x(M“) c Sm’1(r) . But spherical submanifold cannot be minimal in ambient Euclidean

space , therefore we have a contradiction .
2°. If 0k = 0 , then one eigenvalue, say 1,1 , must be zero. If k 2 2 we conclude from

(2.14) that also ab] = 0 . That implies that another eigenvalue, say 312 , is zero which is
a contradiction since 1,1 and htz are two different eigenvalues. If k = 1 (and 61 = 0)
then 'x' = 20 + '12", with A? =0,sobytakingtrace,0=tr(Af =-2n again

contradiction. We conclude that i cannot be finite type map. 9

46

If x : M“ -) E’“ is spherical submanifold, i.e. submanifold of the unit
hypersphere centered at the origin, then 'x' is also an isometric immersion by virtue of
2*(X) = X xt + x Xt . It is interesting to see how certain properties of the immersion x are

reflected in the immersion i and vice versa. To that end we prove the following

Theorem 2.5. Let x: M“ -) S’“'1 C E'“ be an isometric immersion and let
i : M“ —) SM(m) be its quadric representation. Symbols with ~ are related to the
immersion 'x' , those without ~ to the immersion into E'“ and symbols with ' relate to
the immersion into 8“"1 . Then
i) llh'll =const (=9 llhll=const

IIH ll =const :9 "H" =const

ii) M“ is pseudoumbilical in SM(m) via '1? a M“ is pseudoumbilical in E“1 via x.
iii) 15 H = 0 a: h'=0 , i.e. M“ is totally geodesicin 8““l .
iv) PE =0 e» h'=0.

Proof. i) Since 32' isanisomeuic immersion we have A3? = - n H , and using

(2.3) we get
~ 2
(2.15) H: mxt+xH‘)+H;eie§ .

Using the fact that H = H' - x , we obtain

2

~ ~~ ~ ~ . 2 2:2
IIHI|2=g(H,H)= %tr(Hz) = ||H||2+ 2+ 3 = IIHII +

This proves second equivalence of i ) ( cf. [C 4], Lemma 4.6.4 , p. 152 ). The first

equivalence can be proved using similar reasoning. In fact

47

E (x, Y) = 53*(h'(X, Y)) + XY‘ + Yxt - 2< x, Y > rott ,

andhence ME 112: llh ||2+n2+2n.
ii ) We differentiate (2.15) along vector field X 6 TM to get

fol' = VX(Hx‘+xH‘) + ggvxaieg)
= _[(AHX)x‘+ x(AHX)t] + [(DXH)x‘+ x(DXI-DI]

+ (HX‘i-XH‘) + % 2[er(ArX)‘+(ArX)e:]
r=n+1

We simplify this by choosing x = em , and observing that H = H‘ - x we obtain

n+2 ~

VXH = — [ xr<AHX) + — x.(X)1
(2.16) + [(DXH') x‘+ x (DXH')‘] + (H'x‘+XH")
2 m-l
+ a 2[ He(Al)()‘+(ArX)e] .
f=n+
On the other hand, VXH = - AfiX + DXH , so by comparing components of

(2.16) which are tangent and normal to M we obtain

(2.17) Kgx = AHX + “—+2 x

(2.18) fixfi = [(DXH') x‘+ x(DxH')‘]

+ (H'x‘+XH") +§ i[er(Arx)‘+(AYX)e;] .

P—‘T‘H‘

48

The equation (2.17) proves part ii ). Note that in (2.18) the first line of the right hand side

is the component tangent to the sphere 8"“1 via i and the second line represents the

component of DXH which is normal to 8""1 . Therefore, if DXH = 0 , we see that

D'H' =0 and
(2.19) (H'X‘+XH") + % r:2;‘:1[e,(A,X)‘+ (A,X)e;] = 0

Using H' = ii :5“ A,)e, and substituting into (2.19) we obtain

(2.20) % nil{e,[(trA,)X+2(ArX)]‘+[(trA,)X+2(ArX)] cg} = 0 ,

r=n+1

for every X 6 TM . From here we have (tr A,) I + 2 Ar = 0 and taking trace of this
relation we get (11 + 2)tr Ar = 0 , i.e. tr Af = 0 . Putting this back into the relation we
conclude Ar = 0 for every r = n+1, , m—l , or equivalently h' = 0 , which means
that M“ is totally geodesic in 8“” . Conversely, if h' = 0 , then 'x' immerses M as a
minimal submanifold of a hypersphere of SM(m) centered at I/m (V eronese submanifold)
sothat H u(i‘t- I/m)andthereforeDH = 0proving iii).

Part iv) follows from iii ) because V E = 0 implies 1‘5 H = 0. Namely,

0 = Z (Vzh)(ei,ei) = Z 52 H(cpci) " 2 Z H(Vlei’ci)

= n 6211 - 2 Z m{i(z)h(cjrci) ’
1.]

and the second sum is equal to zero.

49

Finally, let us compute second iterated Laplacian of the quadric representation
because it sets the stage for the investigation in Chapter 3 .

Recall that we computed

i

(2.3) Art” = -n(Hx‘+xH‘) - 22eiet .
1

To find A23? we first find - 2 Meet) and then A(H xl + x H) . We can assume
1

that at given point p we have (Vekej )(p) = 0 (normal coordinate system). Then the
Laplacian becomes Af = - 2 ekekf at point p , so we have first
It

Zvekkieit) = Z [h(ekiei) Cit + e, h(ek,ei)t] ,

and then atp
- 24(614) = EVquktecE»
i k,i
= - Z {IAh(ek,ei)°k] ei+eiIAh(ek.et)°k]t}
k.i
(2.21) + z ((1),,kh(ck,e,)]c{+ei [Dckh(ek,ei)]‘}
k.i

+ 2 2 h(ek,e,)h(e,,e,)‘
1t,i
Now we compute each sum separately, first

2 2 h(ek,ei)h(ek,ei)t = 2 z g(h(ckrei)9 r) g(h(ck’ci)’cs) etc:
k,i

k,l,r,s

50

2 g g g(Arcirck)g(AgCi.Ck) ere:

2 2 z g(Arci’ Asci) ere:
r s 1

2 Erma.) crc:

2 tr (ArAchrc: + 6.6:]

then

k2 {[Ah(ekvci)ek] Cit + Cl [Ah(Ck,Ci)ek]t}

E [g(Arck’ei) (Arctic: + g(Arek’ci) ei(Arek)t]

2 £5; (Ardent .

and fmaly, using the Codazzi equation ,

2 {[Dekh(ek.ci)le§ + e. [Dekh(ck.ci)l‘}
k.i

= Z [[(Vckh)(ek,ei)] cit + ei [ (vckh)(cktci)]t}
i,k

51

z {Warming} c: + 9,1(Vcih)(er.er)]‘l
l,k

2 [[Deih(er.ek)le§ + e. {Dartmoor}
k,i

nzuocinnf + ei(DciH)‘]

Substituting these formulas into (2.21) and putting it together we see that at point p the

following equation holds
.. 24(6191) = n 2[(DeiH) of + c,(DeiH)‘l
l i
(2.22) + 2 tr (ArArflchE + Csei]
r.s

_ 2 l
E, (Aromas

Neither left hand side nor right hand side of (2.22) depend on the adapted frame chosen,

so the formula is true for any (local) frame at any point of M .
Next we compute A(H xt + x Ht ) using product formula for the Laplacian

A(H x‘ + x H‘) = [(AH) x‘ + x(AH)t] + [H(Ax)‘ +(Ax)H‘]

— 2 Z [(VeiHXVeiX)‘ + (VeiXXVeiPD‘l

52

(2.23) = [(AH) x‘ + x(AI-Dt] - 2n HHt

+ 2 Z [(AHei)eit + ei (AHei)‘]

— 2 Z [(DeiH) e; + ei (DciH)‘]
i
combining (2.22), (2.23) and (2.3) we finaly obtain the following formula for A2 r .

A25: = - 11 [(AH) x‘+x(AH)‘] + 2n2 HHt

- 2n 2 [(AHei)eit + ei (AHe,)‘]
(2.24) + 4n 2 [(DciH) e; + ei(DeiH)‘]

+ 2 2 t1' (A,A,)re.e:+ e.e:1 - 4 ; (Aroma)

Expression (2.24) can be further broken down into components using the following

formula for AH due to B. Y. Chen ([C 4], p. 271)
(2.25) AH = ADH + 11 A“, 1le + a (H) + tr (VAR) .

m
where en“ ll H , AD is the Laplacian of the normal bundle, a (H) = 2 tr(AHAr)er is so

r=n+2

called allied mean curvature vector and u' (VAH) = tr (VAH) + U" (ADH ).

CHAPTER 3

SPHERICAL HYPERSURFACES WHICH ARE OF LOW TYPE
VIA THE SECOND STANDARD IMMERSION OF THE SPHERE

In the previous chapter we classified submanifolds of a Euclidean space whose
quadric representation is of 1 - type as those which are totally geodesic in a hypersphere
centered at the origin. Of course the same is true when a submanifold is assumed from the
outset to be a hypersurface of the sphere. In this chapter we consider a hypersurface of the
unit sphere centered at the origin (henceforth called spherical) and study those which are of
2 - or 3 - type via the second standard immersion of the sphere. Throughout, we generally
assume that the dimension of a submanifold is greater than one. For finite type curves in

general see [C 4], [C-D-V] ~

1. Spherical hypersurfaces which are of 2 - type via §

In Chapter 2 we derived the formula for the second iterated Laplacian of the quadric
representation 32' of a submanifold M“ (formula (2.24)). If M“ is spherical submanifold,

then the Laplacian of the mean curvature vector H of M“ in E‘“ can be computed as ([C 4],

Lemma 6.4.2 , p. 273 )

53

54

(3.1.1) AH: AD'H' + a’(H') + u(VAH) + (llAgll2+n)H' — norzx

9

where, as usual, symbols with ’ denote objects and quantities related to the immersion of
M“ into the hypersphere S“H , thus, AD. is the Laplacian of the normal connection of M in
S““‘1 , and a'(H') is the allied mean curvature vector in S““‘1 . The mean curvature of M“ in
E“1 is denoted by a and the one in S““'1 by a' . They are related via 012 = a'2 + 1 . g is a

local unit normal vector field of M in 8““1 such that § ll H' , hence H' = at . We also

have
u(VAH) = u(VAH) + u(ADH)
n 11
(3.1.2) = zweiAHki + ZADCiH ei .
1= i=1

where e1, , en is a local orthonormal frame of tangent vectors of M“ .
If M is now spherical hypersurface, then AD'H' = (Aor')§ , a'(H') = 0 and by one
result of [C 6] ( see also [C 5], p. 21 ) we have

(3.1.3) tr (VAH) = n ot'Vot' + 2 A(Vot')

Putting this back into (3.1.1) and combining with (2.24) we have
A251” = — n [Aor' + a'( ”All2 + 3n + 4)] (fix‘+ xgt) — n ( Wxt + xW‘)

+ 2n (not2 + n + 2) xxt + 2 (n20l'2 + 2||A||2) flit
(3.1.4) + 4n [ g(Vot')t + (Va'fli‘l — 4(n+1)2:<==ieit

— 2n a'Z [(Aei)eit+ei(Aei)t] — 4 2 (Aei)(Aei)t ,

55

where A=A§ and w = u(VAH) = not'Vot' + 2A(Va') .
Suppose that 'x' = xxt is of 2 - type. Then we have 'x' = 360 + 35p + iq and hence

2~ ~ ~ ~ _
(3.1.5) A x - (hp+)uq)Ax+ APXqU—xo) — 0 .

Ill order to eliminate constant vector to from this equation we find the directional derivative

VX of (3.1.5) . First, using straightforward but long computation we obtain

VX(A2')E) = - n<X,Vp +AW+4Vot'>(§x‘+x§t)
+ 2n < X , W + 2na'Vor') xxt
+ < x ,4n2a'Va' + 4 VIIAIIZ + 8n A(Vor‘)> at}
+ 2(n2a2+n2 +4n+2)(xitt + xx‘)
+ n (p + 4 an (AX)xt + x(AX)‘]

4[(A2X)x‘ + x(A2X)t] - n [ (waptt + x(wam

+

- n9< Xt‘ + 4X5 + 4n [( Vx(Voc'))§t + E.(Vx(Va'))‘l
(3.1.6)
— 2(n2a'2 +2 llAll2+2n+2)[ (AX)? + E_,(AX)t]
4n a'1(A2X)E.‘ + §<A2xt1 — 41 (Aimt + t<A3X)‘1

n (xwt + wx‘) — 4n [ (AX)(V0t')t + (Vot')(AX)‘]

— 2n < x ,Va'> 2 (peeps,t + c,(Ae,)‘]
i

56

- 2n 01' 2 [((VXA)ei)ei‘+ei((VxA)€,)t]

- 2 Z [((VxAzkth+ei((VxA2)ei)‘] ,

where 11A112= trA2 , and p = Aa' + ot'(11A112 + 3n+ 4).Also, we easily obtain

(3.1.7) in = Vx(xxt) = th+ xxt

(3.1.8) fixmi) = 2(n+l)(th-1- xx‘) - n<x,Vot'>(§x‘+xE})
+ not'[(AX)x‘+ x(AX)‘] — not'(§x‘+X§‘)
- 21m»:t + §<AX)‘1 .

Denote the left hand side of (3.1.5) by Q( i ) , i.e.

(3.1.9) Q(ii') = A2; - (2p+).q)Ai‘t'+ Apxq(i'-it'0) .

From g ( VX[Q( 'x' )] , xxt) = 0 using (3.1.6) - (3.1.8) we obtain W + 2n0l'V0t' = 0

and combining with (3.1.3) we have
0 3 l O
(3.1.10) A(Va)= - 2 nor V01 ,
and therefore (3.1.3) yields
(3.1.11) W= u(VAH) = — 2n a'Vot' .

From g(Vx[Q(i't')], gg‘) = 0 , we get 4n20t'V01'-1- 4VllAl|2+ 8n A(Vot')=0,

and therefore, using (3.1.10) , we have

57

(3.1.12) V 114.112 = 2n2 a'Va' ,

or IIAII2 = n2 01'2 - c (c is a constant), which implies that the scalar curvature of M is
constant , since 112 (2'2 - "A"2 = 1: - n(n+1) by the Gauss equation .

Let U = [ p e M I V(0t')2 at 0 at p I. Then U is an open (possibly empty) subset
of M, and on U we obviously have also 01' at 0 and Vor' at 0 . If U is nonempty , then

by (3.1.10) we see that Vor' is an eigenvector of the shape Operator A on U with the

eigenvalue - % nor' . Now on U we choose unit tangent vector el to be in the direction of
Va' , i.e. c1 = Vor'/IIVOt'l| . We find ele: component of VX1002 )] on U setting first

X = Vor' . Combining (3.1.6) , (3.1.10) and (3.1.11) we get the following by
exploiting thXIQ(8)1.e,e{) = o

0 = 16n20t' ‘g‘((Va‘)(Vot')‘, ele‘l) - 2n|lVa'|l2< Ae1,e1>
- 2n u' < (waiter:l ,e1> - 2 < (VV(,,..4>.2)el ,e1>
= 8 n201'llVor'Il2 + 3 n201'IlV0t'll2
— Zn or' < Vvav(Ae1) - A(VVa'cl) , e1>
- 2 < Vvat(A2el) — A2(Vvave1) , e1> .

Note that <VVo‘te1 , el > = 0 , and therefore also < A(VVa'el) , el > = 0 and

< A2(VVot'°l) , e1 > = 0 . Hence, the calculation above continues as

0 = llnza'IIVa'llz+ 3n2a'(Va')(a') - g-HZCVQ'XG'Z) = 5n2a'11Vot'112.

From this we conclude or' = 0 or Vor' = 0 at any point of U. However, 11118 18 a

contradiction, and hence U must be empty. This means that V(or')2 = 0 everywhere on

58

M , i.e. a' = const . Therefore, a hypersurface of 8“"1 which is of 2 - type via '56 must
have constant mean curvature a.‘ in sphere .

Let us remark that in order to find different components of VX[Q( SE )] it is not

absolutely necessary to use long formula (3.1.6). We can also find those components
indirectly, for example, §§‘- component can be found in the following way . Let

Q'(i') = A23? — (xp+xq)Ai'+ prq 32' . Then

0 = <Vx[Q('i)]. §§‘>

X< Q'( I).§§‘> - <Q'<‘i). We?»

X< Q'( i).§§t> + < Q'(i').(AX)§t +§(AX)‘>

= X(nza‘2+2|lA||2) + 4n<V0t',AX>

= <x, 2n20t'Va' + 2V|| All2 + 4n A(Va')> ,

so that nza'Va' + V "A "2 + 2n A(Va') = o as before.Similar1y for xxt- and

(V a')(V ()t')t - component .

We are ready now to prove the following classification result

Theorem 3.1.1. Let x : Mn —-) Sn+1 be an isometric immersion of a compact
n - dimensional Riemannian manifold M into 8"“1 (n 2 2). Then 52' = xxt is of 2 - type
if and only if either

(1) M is a small hypersphere of Sn+1 of radius r < l , or
(2) M = Sp(rl) x Sn'p(r2) , with the following possibilities for the radii r1 and r2 :

_ .. +2 2 n- __}3_ 2:31:22

59

The immersions in (l) and (2) are given in a natural way .

Proof. If M is one of the submanifolds described in (1) and (2) , then M is of 2 -
type via the second standard immersion of the sphere as shown in [BC] . Conversely, let
us assume that for a spherical hypersurface x : Mn —) Sn+1 the quadric representation i
is of 2 - type. Then (3.1.5) holds, and from the above we see that the mean curvature or' of
x is constant. In that case Va' = w = V IIAIIZ = Vp = o , and the formula (3.1.6)

simplifies, so that the part of (3. 1.6) which is tangent to M reduces to
2(n2a2-t-n2 +4n+2)(xttt + xx‘)
+ n (p + 4 a’)[ (AX)Xt + x(AX)t]

+ 4[ (A2X)x‘ + x(AZX)‘] ,

where p = a'(llAll2 + 3n+ 4) isconstant.Let ek,k= 1,2, ...,nbealocal
orthonormal vector fields which are eigenvectors of A (principal directions) and let 11k be

the corresponding principal curvatures . We set X = ck in (3.1.6) and compute the

component tangential to M . Then fiom ‘g‘ ( Vek[Q( i )] , xef‘ + ekx‘) = O we obtain

0 = [2(n2a2 +n2+4n +2) - 2(n+ 1)0.p+7.q) + Aplq]
(3.1.13) 2
+ n[p-(7\.p+2tq)0t']ttk + 4ttk.
This is a quadratic equation in uk with constant coefficients which do not depend on k and
the equation is not trivial ( O = O ) because of the term 4 it: . We conclude, therefore, that
each principal curvature is constant and that there are at most two distinct principal
curvatures . If M has only one principal curvature, i.e. if it is umbilical, then M is a small

hypersphere of S'”1 . If M has two distinct (constant) principal curvatures then M is the
Standard product of two spheres, M = Sp(r1)x Sn'p(r2) with rf 4» r3 = 1 (see [Car 2], or

60

[Ry] ) . Then, according to [B-C] (Lemma 3), such product will be of 2 - type via 32' if and
only if the radii satisfy precisely those three possibilities listed in (2) 0
Theorem 3.1.1 is a generalization of a result of M. Barros and B.Y. Chen, who

proved a similar theorem assuming M to be mass - symetric (cf. [B-C], Theorem 3 ) .

2. Minimal spherical hypersurfaces which are of 3 - type

and mass - symmetric via i'

Since computations for the third iterated Laplacian of i? become more involved and
considerably more difficult to handle we restrict our investigation to minimal spherical
hypersurfaces which are of 3 - type and mass - symmetric via 52' . For minimal

hypersurfaces in sphere, calculations fi'om before give

(3.2.1) A3? = A(xxt) = 2n xxt - 224.21g!
(3.2.2) — A (Eat-1 )..-.= 2n sott +2 "All2 :6 — 22¢; - 2 z; (AekXAek)t
(3.2.3) A(ggt) = 2 ||A||2§§‘ — 2 g(ActXAct)‘

(3.2.4) A2)? = 4n(n +1) xx‘ + 4 ”All2 gg‘ - 4(n + 1) Zoo} - 4; (Aek)(Aek)t .

We also have the following lemma, which can be proved by direct computation in a

similar fashion as was done to prove formula (2.22) .

Lemma 3.2.1. If e1, , en is a local orthonormal basis of tangent vector fields of

61
n
M , and AA = Z [ vaei A - Vei(veiA)] is the trace Laplacian of the shape
i=1
operator, then
- A { 2(Ac,)<Ac,)‘} = — 2 [((AA>c,)(Ae,)‘ + (Ac,)«AA>e,)‘1
1 1

— 2 X(AeiXAei)‘ - 2 2(l«>.2t:i)(.¢tzoi)t

+ 2(n A2)xx‘ + 2(n A4)§§‘ — 2 (tr A3)(x§‘+ ax‘)
(3.2.5)
- 2[(ttVA2)xt + x (trVA2)‘]

+ [2 (WM) -A2(trVA)] g‘ + E,[2(trVA3) -A2(trVA)]t

+ 2 2;, [(VckAkilflVekAkil‘

Each sum here is independent of the frame {ei} chosen .

One of the results of K. Nomizu and B. Smyth in [NS] is computation of AA for

spherical hypersurface with tr A = const . Namely,

(3.2.6) AA=(trA2-n)A + (trA)I - (trA)A2 .

Because we assume tr A = 0 we will have

(3.2.7) AA = (tr A2 - n) A

Now taking Laplacian of (3.2.4) and taking into account (3.2.1 - 3) and (3.2.7) we get the

following formula for A3 i' .

A3: = 8[n(n+l)2+ trA2]xxl

62

+ 4[A(trA2) +2 (tr'A2)2+2(n+1)trA2 + 2 nA‘]§§‘

— 8 (tr A3)(x§t + gx‘)
(3.2.8)
— 4 [ (V(tr A2))xt + x(V(tr A2))‘]

+ [ §V(tr A3) + 12 A (V(tr A2))] at + g[ §V(tt A3) + 12 A (V(n A2))]t

— 8 (n + 1)2 Zoe; — 16 (1 + tr A2) Z(A«:,)(Ati)t - 8 2(A2ei)(A2ei)t

+ 8 2.; [(VakAkilKVekAkilt

Each sum in this formula is independent of the frame chosen.

Suppose now that Mn is mass - symmetric and of 3 - type via '1? so that 3‘60 = BIL—2

and
3» 2» ~ t _I__ -
(3.2.9) Ax+an+be+c(xx-n+2)—0.

where a, b and c are constants.(They are equal to elementary symmetric functions of three

eigenvalues of the Laplacian which arise from the decomposition i = i0 + xp + xq + it . )

Using (3.2.1), (3.2.4) and (3.2.8) we find different components of (3.2.9) such as xxt

component, 5? component, x? + §xt component etc. For example, comparing xi‘ + éx‘

components of left and right hand side of (3.2.9) we see easily that tr A3 = 0 . Comparing

xxt components in (3.2.9) we obtain

8[n(n+1)2+ trAz] +4an(n+1) + 2bn+ “LB—:3),

and consequently tr A2 = const . Similarly, from £5} component of (3.2.9) we have

63

4[A(trA2) +2 (trA2)2+2(n+1)trA2+ 2 ttA4]+4auAu2 -c 3372 o,

and hence tr A4 = const as well. We conclude,therefore, that for the dimension n S 4
minimal spherical hypersurface which is of 3 - type and mass — symmetric via i must be
isoparametric, i.e. its principal curvatures must be constant.

Because of this obvious importance of isoparametric spherical hypersurfaces for

our investigation we consider next some examples.
Example 3.2.1. Cartan hypersurface

According to Cartan theory ( [Car 2, 3] ), there is only one (up to congruences of
the sphere) compact minimal isoparametric hypersurface M3 of S4 with three principal
curvatures. This hypersurface is a tube about Veronese surface and is usually called the

Cartan hyperswface. It is a homogeneous space of type SO(3)/Z2 x 2,2 and an algebraic

manifold whose equation is
2x;3 +3(xf+x§)x5 —6(x§+x})x5+ 343(x12- x§)x4 +6\/—3x1x2x3 = 2 ,

with Xx? = 1 ( see Ch.1, Sect.4 ). The Cartan hypersurface has three distinct principal
curvatulres k1 = - ‘13 , k2 = 0 and k3 = V3 , hence by the Gauss equation the scalar

curvature is equal to 0 .
We are now going to show that the Cartan hypersurface M3 = SO(3)/Z2 x Z2 is an

example of minimal spherical hypersurface which is of 3 - type and mass-symmetric via 32'.
Let {ei}, i = 1, 2, 3 be an orthonormal basis of principal directions. Then, for the Cartan

hypersurface, equations (3.2.1) and (3.2.4) become respectively
(3.2.10) Ai = 6 xx‘ — 2 ( clef + eze; + e3eg)

(32.11) A25? = 48 xxt +24§§t - 12(ele:+ e3e§)- 16(e1e:+ e2e§+ eseg).

64

In [Car 2], Cartan also computed the connection of M3 , namely

3_1 1__1 2_ 3
m2--m,m3-2a)2andcol—-m ,

where the connection forms are computed with respect to the basis [ei} of principal

directions. Substituting this into (3.2.8) we obtain
(3.2.12) A33: = 432 xx‘+ 624§§‘ - 408(eleg + e3eg) -80(ele§ + e2e5+ e3e; )

From Lemma 1.3.1 we have I = xxt + :6 + ele: + ezeé + e3e§ , and combining with

(3.2.10 - 12) we have

(3.2.13) A3i—34A23r' +328Ax—960(x-§)=0.
It follows that the Cartan hypersurface is mass - symmetric and of 3 - type via i since it
cannot be of l- or 2 - type by the classification in Section 1 of this chapter. Moreover, we
easily find the three eigenvalues determining the order to be AP = 6 , Aq = 8 , A, = 20 .
As a byproduct, we found three eigenvalues of A for the Cartan hypersurface. As a matter
of fact, the spectrum of the Cartan hypersurface was computed in [M-O-U] , from which

we determine its order via 32' to be [2, 3. 3] .

Note also that, according to [H-L], the Cartan hypersurface arises from the isotropy
representation of the symmetric space of rank two which in this case is SU(3)/SO(3) .
Namely, the Lie algebra su(3) decomposes into a direct sum of the subalgebra so(3)
and the vector space m which is identified with the set of 3 x 3 real symmetric matrices
withzerotrace, m = [ iAl Ae SM(3), trA=O }.Using <X,Y>= -tr(XY) asan

inner product on su(3) , this decomposition is orthogonal, and 80(3) acts isometrically on

65

the Euclidean 5 - space m by inner automorphisms (see [F], [Ko], [M-O-U], [Ce—Ry], p.
298 ). The Cartan hypersurface is the orbit of the point

iOO
x=71§ O-iO em
000

Example 3.2.2. Minimal isoparametric hypersurface in 85

with 4 principal curvatures

As discussed in Section 4 of Chapter I , there is only one minimal isoparametric

hypersurface M4 in $5 with four cm'vatures; it is the image of the following map

SI X 832—) SSC E6

(3.2.14) (9 , (x,y)) —> z = ei6 (cos t x + i sin t y) ,

for t = 1t/8 . In general, (3.2.14) defines the isoparametric family studied by Cartan [Car 3]
and Nomizu [N 1-2] . It is an algebraic family defined as [Car 3]

2
cos4t == (xf+x§ +...+x§)2- 2(xg -x§-2x1x5+2x2x6)

2
- 2 (2x3x4 - 2x1x6 — 2x2x5)2 , 2 xi = 1

To parametrize the Stiefel manifold 83,2 choose x to be an arbitrary vector of the
sphere 52 , i. e. x = ( cosa cosB , cosoc sinB , sina) , and choose vectors u and v of S2
that span the plane perpendicular to x , e.g. u = ( ~sinB , cosB , O) and v = u x x ,
thus v = ( sina cosB , sina sinB , - cosa ) . For any vector y J. x, y = cos¢ u + sin¢ v .

80 y = ( -sinB cos¢ + sina cosB sintb , cosB cos¢ + sina sinB sin¢ , - cosot sin¢)

66

Denote r = cos t and s = sin t . Then fiom (3.2.14) and the consideration above we have

the following parametrization of M4

21 = r cosO cosot cosB - s sine (- sinB coso + sina cosB sin¢)
22 = r cosG cosa sinB - s sine (cosB cos¢ + sina sinB sin¢)
2:, = r cosO sina + s sine cosa sin¢

(3.2.15) z4 = r sine cosa cosB + s cosG (- sinB cos¢ + sina cosB sin¢)
25 = r sine coson sinB + s c036 (cosB cos¢ + sina sinB sin¢)

z6 = r sine sina - s cose cosa sin¢

We differentiatez toget basis vectorfields 81:2- , 32: j- , 83 =2- , 84 = i as
89 8a BB 8(1)
follows
8 . . . .
g = ( - r s1n6 cosa cosB - s COSe (- 811113 005$ + 51““ 0055 51114)) .
- r sine cosot sinB — s cosG (cosB cos¢ + sina sinB sin¢) ,
- r sine sina + s cosG cosa sin¢ ,
(3.2.16) r cosO cosa cosB - s sine (- sinfl cosq) + sina cosB sin¢) ,

r cosG cosoc sinB - s sine (cosB cos¢ + sina sinB sin¢) ,

r 0059 sina + s sine cosa sin¢ ) .

NOte “13138—6 =(-z4’-25’-z6’ 213 Z29 Z3)

~53— : ( — cosB (r cosG sina + s sine cosoc sin¢) ,
0t — sinB ( r cosB sina + s sinO cosa sin¢) .
r cos9 cosa - s sinO sina sinq) ,
(3.2.17) cosfl (- r sine sina + s cos6 cosa sin¢) ,

sinB ( - r sine sina + s c059 cosa sin¢) ,

67

r sine coso: - s cosO sina sinq) ) .

313 = ( sinB (- r cosO cosa + s sine sina sin¢) + s sine cosB cos¢ ,
— cosB ( - r cosO cosa + s sinO sina sin¢) + s sine sinB cosq) , O ,
(3.2.18) — sinB ( r sine cosa + s cosO sina sin¢) - s cosO cosB cosq) ,

cosB ( r sine cosa + s cose sina sin¢) - s cosO sinB cosq) , O )

i = ( - s sine ( sinB sinq) + sina cosB cos¢) ,
- s sine (- cosB sinq) + sina sinB coso) ,
s sine cosot cos¢ ,
(3.2.19) s cosO ( sinB sinq) + sino: cosB cos¢) ,

s cosO (— cosB sin¢ + sina sinB cos¢) ,

- s 0080 00511 cos¢ )

We compute componenets of the metric tensor as gij = < 3i, 8]. > to get the following

matrix G = ( gij ) of the metric tensor

( l 2rs sintp -2rs cosa cost]: 0 \

2rs sin¢ r2+s2 sin2¢ -szcosa cos¢ sin¢ 0

. 2 2 . 2 2 -
-2rs cosa cos4t -32 cosa cos¢ srn¢ 52+ cosza (r - s srn 4)) -s srna

K 0 0 -s2 sina 82 )

 

 

68

The determinant of this matrix is computed to be det G = r2s2(l - 4r282)C082a . We find
the inverse matrix of G to be 0‘1 = 3ng B , i.e. gij = (1.716- bij , where B = (bij) is symmetric

matrix with the following entries

11 = rzszcosza , b12 = b21 = - 213s3sin¢ cosza , bl3 = b31 = 21'3 s3cos0t cos¢ ,
b14=b41= 2r3s3sin0t cosa cos4t, ha:2 3 cos 2a [r2 + s2(1- 41'2)cos2 4)],
b23 = b32 = 4(1 - 412) cosa cosq) sin¢ , b24 = b42 = $40 - 4r2) sina cosot cos¢ sinq) ,
33 = 2(9 + 52(1 - 4r2)sin2¢] .1134 = 1:43 = szsina [:1 + s2(1 - 412)sin2¢] ,

b44= 52(1- 412.62) + s4(413- 1) cos2¢ sinza + (11 — $2)(1- 4&2) cosza .

Next we compute Christoffel's symbols. Nonzero ones are given as follows

2 2 r2 2
- . 1 -s
11 - 1 ___L_2_%_rsrrss srnacosq) , 54 = 42 = —'(—2_%;S-4rs coscp ,

23- 32- 1_4

2 2
- g 0 1 1 rs r - S .
1 _. - 47:12” S srna cosa srn¢ , 34 = 43 = J—gfl s cosa srn¢ ,

33- 1-4rs 1-

s
r43..- r3- —;sinacos¢ , I24: 131:; cost ,
2 2
—si-(r-:—'§’-%‘sin¢cos¢,
- rs

Fz3= I32: 84—3—3 sina sin¢ cos¢ , P = 1'32

1-4:

2 - . s2r2-s -2
I33 = sinacosa[1+ 21%??srnzo] , 1134:1713: — 4—31-4” srnocosot ,

 

s Sin¢
m3=r§1=-;tanasin¢ n4=r§t1=§ .

2 2 52,1- c052¢
P =r3.=-anatt+ 3:71.03“ B4=132=473igsg

69

2+2
l"3 = _ 28 -s . . - £0.15;- _
33 1-4! s srna s1n¢ cos¢ , P -1‘33= I-“ 8 51nd) cosq) ,

 

P12 = P2”: --rs- cosq) , 1113 = F31 = - Sim [s2+(r2-s2)cos20t] ,

rs cosa
F14 = H1=§masin¢, F422 =- sin¢cos¢,

F4 =11?” =-co;sa[sin2¢ + #51?st acoszo],

2 2
==I4'2 473%— tanacoszo, 33=sin¢cos¢[1— ——2—2-1r4s cosZa] ,
- - 1'8

2 2
F544 = :3 = Life} sina sino cos¢

l-4rs

We want to find the shape operator A of the hypersurface and the basis of principal

directions . But first we need to find the unit normal direction § . It turns out that g is

obtained by differentiating z with respect to t , i.e. take = —-g- . So we get
I

§ = ( cosB (s 0050 cosa + r sine sina sin¢) - r sine sinB cos¢ ,
sinB (s 0030 cosa + r sine sina sin¢) + r sine cosB cos¢ ,
s cose sina - r sine cosot sin¢ ,
(3.2.20) cosB (s sine cosa - r cosO sina sintp) + r 0050 sinB cos¢ ,
sinB (s sine cosa - r cosO sina sin¢) — r cosO cosB cos¢ ,

s sine sina + r cosO cosa sin¢ ) .

For every i, j = 1,2,3,4 we can compute < A(ai). aj > = - < 73;. 8,- > and find the

matrix of A in the basis {8i} . We get

70

cost)

 

r2_ 2
M91) = W {—2rs8l + sinoa, - a3 — tana cos¢84} ,

COS“

2r-
A(82) = -—2:1-{ sinoa + -(cos2 ¢- 2r2)<')2

sin¢ cos¢ a

+ -tana sinq) cos 84
cosa 3 2 }

,s
1'
1'2-82

A(83) = -l-—4r2_s_2- {— cosa cos¢ 81 + geosa sin¢ cosq) 82

+ 15, (sin2q) — 2r2)83 - }lgsinafl'z ' 52 sin2¢) a4 }
A(a4) =2 3

Note that even though A is symmetric operator, the matrix of A in this basis is not

symmetric since {ail is not an orthonormal basis . Minimal hypersurface in the family

V2+‘/2 - 1t V245
2 —— .

,s=srn- = 2

 

00

(3.2.14) is obtained when t = 1t/8. In that case r = cos g =

Principal curvatures of minimal M4 are given as follows :

k1=~12+1,k2=-s/2-1,k3=\/2-1,k4=1-x/2

That follows from the Cartan's identity (1.5.1) or Mi'nzner's Theorem 1.5.2 . Next we

find the orthonormal basis of principal directions by diagonalizing matrix of A in the basis

{Bi} . We get the following principal directions corresponding respectively to the curvatures

k1’k21 1‘3, k

71

 

 

V4+2312 8 . a cos¢ a a

62 — 2 {39 sumac + cosot BB + ma cost!) 5
44-242 . cosoa 3

c3 = 2 {‘36 " 81114152; '1' E g + tanor 008(1) 3 }

a sin¢ a . 8
e4 ‘1 V { cost!) 3a 4» sa 313 + tana sm¢ }

To check if M4 is of 3 - type via i' or not we find connection coefficients with respect to the

basis {ei}. For example, we compute

(02k?) = < Vc3el , e2 > = 0 , (1)2(e4) =V2-«f2 ,

But combining the equations of Gauss, Codazzi and condition (3) of Theorem 3.2.1 below

it follows that in order that M4 be mass - symmetric and of 3 - type via 2 we must have

1m,<e,)1= [m,(e.)12=22"2 .and imi<e1>1= toa§<e2>1=2§r2

Therefore, M4 is not mass - symmetric and of 3 - type via a: .

Now we prove the following characterization of minimal spherical hypersurfaces

which are mass - symmetric and of3 - type via a: .

Theorem 3.2.1 Let x : Mrl -) Sn+1 be an isometric immersion of a compact
manifold Mn as a minimal hypersurface of Sn+1 . If 'i' is mass - symmetric and of 3 - type
then
(1) trA =nA3=0,

(2) tr A2 and tr A4 are constant ,

(3) u(va)2= <A2X,A2X>+ p<Ax,Ax>+ q<X,X>, Xe TM

2 4
where p and q are constants (depending on the order of M , tr A and tr A ) .

72

Conversely, if (1), (2) and (3) hold then M is mass - symmetric and of 1 - , 2 - , or 3 - type
via i .
Proof. Suppose that i' is mass - symmetric and of 3 - type so that (3.2.9) holds.

As before, from x§t + §xt component of (3.2.9) we get tr A3 = 0 , and xxt and £5}

components give respectively
(3.2.21) 8[n(n+1)2+ trA2] + 4an(n+ 1) + 2bn + c E = 0,
(3.2.22) 8[(trA2)2 + (n+1)(trA2) + trA4]+ 4a(trA2) -c 31—2 = 0,

Obviously tr A2 and tr A4 are constant, and (3.2.8) simplifies to

A35: = 8[n(n+l)2+ trA2]xxt+ 8[(trA2)2+ (n+1)<trA2)+ mac

(3.2.23) — 8 (n + 1)2 zeta} - 1o (1 + tr A2) 2(Aeieri)‘

- 8 X(Azegmzei)‘ + 8 z [(VekA)ei][(VckA)ei]‘
‘ m:

We readily observe that A3 'x' , A2 if, A a: , I are all normal to hypersphere Sn”1 ( follows
e.g. from the proof of Lemma 1.5.1 ). Next, we find X Yt + Y Xt component of (3.2.9)
for arbitrary pair X , Y of vector fields on M. Observe first that

Z g((VekA)ei, X) g((VckA)ei, Y) = 2 g(ei, (VckA)X) g(ei, (VckA)Y)
i,k i,k
= 2 8((VxA)ct. (VYA)ct)
k

= tr (VxA)o(VyA)

by the Codazzi equation and symmetry of the operator VekA .

73

Now applying g ( — , XYt+ ‘trxt ) to (3.2.9) and taking into account (3.2.1),
(3.2.4) and (3.2.23) we get

— 8(n+1)2<X,Y>—16(1+trA2)<AX,AY> — 8<A2X,A2Y>
+8tr(VxA)o(VYA)- 4a<AX,AY> — 4a(n+1)<X,Y>

— 2b<X,Y> -c 313<X,Y> = 0 , fromwhere
(3.2.24) tr(VXA)o(VyA) = < A2X,A2Y> + p< AX,AY> + q< X,Y>,

where p and q are constants given by

(3.2.25) p = g + 2(1+trA2)

b c
(3.2.26) q (n +1)2+ g(n+ 1) + Z + m .

It is easy to see that (3.2.24) is equivalent, by linearization, to

(3.2.27) tr (VXA)2 = < A2X, A2X > + p < Ax, AX > + q< x, x > ,

for any X 6 TM . Therefore, we proved necessity of the conditions (1), (2), (3) .

Conversely, given (1), (2) and (3), we have to show that we can find constants a, b and c
so that (3.2.9) holds . That boils down to solving the system of the following four
equations (3.2.21), (3.2.22), (3.2.25) and (3.2.26) for a, b, c . This system of four linear

equations in three unknowns can be uniquely solved if the eliminate is zero, i.e.
(3.2.28) trA4+ptrA2+qn+(n-trA2)trA2=0.
But this formula is always satisfied under our conditions (1) - (3), by virtue of

A(tr A2) = tr(AA)A - IIVA 112

NIH

O:

74

(cf.[N-S],p. 369) . Therefore P(A) (it - 760) = o, where P(t) = t3 + a t2 + bt + 0.

Note that M need not be exactly of 3 - type, i.e. can be of l - or 2 - type, for example if
there is a factor F ofP ofdegree 1 or 2 so that P'(A) (32' - 20) = 0 . 0

We now prove the following characterization of the Cartan hypersurface .

Theorem 3.2.2. Let x : Mn —9 Sn+1 be a compact minimal hypersurface of Sn+1
ofdimension n S 5 . Then i’ is mass - symmetric and of3 - type ifand only if n = 3 and
M3 = SO(3)/Zz x Z; is the Cartan hypersurface .

Proof. From the Example 3.2.1. we know that the Cartan hypersurface is mass -
symmetric and of 3 - type via i' . Conversely, suppose that a: is mass — symmetric and of

3 - type. We will show that Mn is necessarily isoparametric. From the computation carried

out before that is already clear for n S 4 . If we compute A(tr A”) we obtain

A(tr Am) = m (tt A2— n )(tr Am)

(3.2.29)
j k
— 22mm...vovc.Avo...voVe.Ao... oA).
i pet: ‘ ’
In particular, for m = 3 we have
(3.2.30) A(tr A3) = 3 (tr A2 — n )( tr A3) — 6 2 tr [(v,,iA)2 o A]
i

Since tr A3 = 0 by Theorem 3.2.1 , we will have ( {oi} is chosen to be the basis of

principal directions )

o = Ztr[(VciA)20A]

75

2 g((ve,A)2Act. ct)
i,k

2 g((VeiA)20~1tc1t)r Ck)

i,k

Z g((VeiAXlkck). (VciA)ek) , since VeiA is symmetric
i k
= E 78k g((VciA)ek, (VciA)ek) , since VeiA is a tensor

= Z 22k g((VckA)ei, (VekA)ci) s by Codazzi equation
i,k

; 2'1: tr (VekA)2

= 2 Aka: + pi: + q) , by condition (3) ofthe Theorem 3.2.1
k

= trA5 + ptrA3+ qtrA

= trA5

Therefore, conditions (1) - (3) of the Theorem 3.2.1 imply also tr A5 = 0 . We conclude
that for n S 5 the hypersurface M has to be isoparametric. If M has only one curvature it
has to be umbilical in Sn+1 and therefore (since it is minimal) great hypersphere which is of
1 - type via 5:. If M has two distinct principal curvatures and is minimal it must be Clifford
minimal hypersurface M = Mpm-p = Sp( 2) x Siva]?! ) ([C 1], pp 87, 97 ). But
the product of spheres that satisfies the conditions of our Theorem 3.2.1 must be of 2 -

type as can be seen from the following argument.

Suppose 2.1 and A2 are the two principal curvatures of multiplicities m1 and m2

respectively , Then trA = u- A3 = 0 imply mlAl + m2).2 = mllz’; + mzhg = 0 . Also, we

76

have 1 + 2.112 = 0 (by e.g. (1.4.1) ) . Using this to eliminate m1, m2 and 12 we obtain
2.2 =Af=q/p.Thus, p=q=nl2 , Al: iland7tz=+1.So, n=p+qhastobe
even, p = n - p , p/n = 1/2 and Sp( 3) x S“'1’(‘\jT'—t;T2 ) = 8901;) x spa/g) . This
hypersurface is mass - symetric and of 2 - type by Lemma 3 (case H) of [B-C] for n = 2p .
If M has three curvatures, then according to the classification of Cartan M is the Cartan
hypersurface which indeed is mass - symmetric and of 3 - type via 32' . If M has four
principal curvatures, then the result of Takagi [T 3] classifies such hypersurface as the one
considered in Example 3.2.2 which is not of 3 - type via 32' . Finaly, M cannot have five
principal curvatures by the result of Miinzner (Theorem 1.4.3). This completes the proof of
the theorem. 0
Remark. The proof above does not a priori exclude the case n = 1 . Actually, if
n = 1 , there are no minimal curves in S2 which are of 3 - type in SM(3) via x" because
such a curve is automatically a great circle of 82 (totally geodesic), and therefore of 1 - type
via '12' . Namely, if x : C —-) S2 is a minimal curve parametrized by the arclength s, we
have Ax=x, i.e. x"+x = 0 andhence x(s)= asins + bcoss , a,be E3. From
<x,x>=1 weget Ia|=lb|=l and <a,b>=0.AsphericalcurveCwith these
properties is the great circle lying in the plane perpendicular to the (constant) vector

x'x x =axb.

Theorem 3.2.2 gives a new characterization of the Cartan hypersurface in terms of
the spectrum of its Laplacian . For other characterizations see [P-T], [T 1], [Ki-Na] .

In dimensions greater than 5 there are other examples of spherical hypersurfaces
which are of 3 - type and mass - symmetric via '12' . In fact every minimal isoparametric
spherical hypersurface with exactly three different principal curvatures is of 3 - type (see
below). It would be interesting to decide if any spherical hypersurface which is of 3 - type

and mass - symmetric via 32' is necessarily isoparametric.

77

Lemma 3.2.2. If M“: Sn+1 is a compact minimal isoparametric hypersurface

which is mass - symmetric and of 3 - type via 32' then M" is necessarily homogeneous with
v = 3 or 4 distinct principal curvatures.
Proof. First, we saw before, from the proof of Theorem 3.2.2, that if v = 1 or 2

then 32' is not of 3 - type . If there are six distinct principal curvatures, then by Theorem

1.4.2 the curvatures I:i have the same multiplicities and they are given as

0029 ’ CONN?) . C°t(9+§) . 601(9 +22") . cot(9 +231) , cot(0+%t

From minimality condition we obtain 0 = 152- and then find curvatures to be (in descending

order)

2+\f3 .1. 2-‘13. -(2-\/—3) . -1. -(2+\/3)

We see that these hypersurfaces satisfy conditions (1) and (2) of Theorem 3.2.1 and to
determine if they are of 3 - type and mass - symmetric via 32' one needs to check the
condition (3) . It is likely (but still not known) that all isoparametric spherical hypersmfaces
with six curvatures are homogeneous. That is proved when m = 1 ( [D-N 1] ), classifying

such hypersurface as G2/SO(4) , but not yet for m = 2 . If v = 4 , then

k1=cot6 , k2=cot(0 +315) , k3=cot(9 +159) , k4=cot (0 +31%),
and there are at most two different multiplicities m1 (of k1 and l23 ) and m2 ( of k2 and k4 ).
Then fiom tr A = 0 and tr A3 = 0 we get respectively

cos 20 sin 20 m
m — = 0 , i.e. tan229 = —-1- ,
1sin 20 m2cos 29 m2

 

 

78

cos20 (4 - sin220) _ m sin20 (3 + sin229)

 

 

 

 

= O ,
1 sin32 9 2 cos329 from where
- 2
El _ 4 3 + srn 29 _ El ,
m2 - tan 20 —-———4 - sin220 . Let r — m2 . Then from these two equations we get
3+sin229 . . . . 4- r 2 4-3r
r—1'2Z--siT226-,whrchrrnplressm220=r+1 hencer=tan20=4r_3.

From the last relation we have r = 1, i.e. m1 = m2 so multiplicities of all four curvatures

areequal.Wealsoget 0=%,andfourcurvaturestobe k1=xl2+1 , k2=xl2 - 1,

k3 = 1 - J2 , k4 = - V2 - 1 . Therefore, as argued in Sect.4 of Ch.1, Theorem 1.4.4 of

Abresch implies that the common multiplicity of curvatures is 1 or 2 . If the common
multiplicity is 1 than M4 has to be the hypersurface considered in Example 3.2.2 which is
not of 3 - type via 32' . If the common multiplicity is 2 , then M8 is minimal homogeneous
hypersurface in 89 of type Sp(2)fl‘2 . In the next lemma we show that all minimal

isoparametric spherical hypersurfaces with v = 3 are indeed of 3 - type via 32' . 0

Lemma 3.2.3. If Mn C Sn+1 is a compact minimal isoparametric spherical
hypersurface with exactly three distinct principal curvatures then Mn is mass - symmetric
and of 3 - type via 32' .

Proof. From (1.1.5) and the Gauss equation (1.1.27) we obtain the following for
principal directions ei , ek and corresponding curvatures Ai , M (i at k)

R(ei,ek,ek,ei)=1+ 2,2,, = e,(to [(ek» - ek(ot;(e,))
+ Z @(cmflet) - 2 (0869033919
1' 1

(3.2.31)
- X «termite» + Z <o£<ct>coi<ep
1' J

79

For an isoparametric hypersurface, the Codazzi equation (VciA)ek = (VckA)ei is

equivalent to the following

(3.2.32) (wk-2.1.) m{(c,) = (xi-2.1.) «g(ck) , for every i, j,k ,
and hence

(3.2.33) 022031,) = o , for Ak=xj¢x,.

Therefore, if If: M formula (3.2.31) reduces to
(3.2.34) 1+ 3.9., = — 2.03.39.03.59 — 2mi(ei)co;(ej) + Zoflemkej) .
j i J

All four minimal isoparametric spherical hypersurfaces with three distinct principal
curvatures have curvatures equal to - «f3 . 0 and V3 , and the common multiplicity m

satisfies m e {1, 2, 4, 8} , so that tr A = tr A3 = O . In order to prove that these
hypersurfaces are of 3 - type and mass - symmetric it is enough to check condition (3) of

Theorem 3.2.1 , which can be also written as

(3.2.35) tr (VeiA)2 = 7»? + p 7»? + q .

where ei is a principal direction , 3., corresponding principal curvature, and p and q

constants. We transform tr (VciA)2 as
(3.2.36) U(VeiA)2 = 1.20““ Apztcoflcinz
-J

Let e1, , em be the set of principal directions that correspond to - V3 eigenvalue,

. . . . . 1 e, and
emu , , e2"1 the set of pnncrpal directions that correspond to O eigenva u

80

em”, , 03m those corresponding to ~13 eigenvalue. We use the boldface type to denote

the following set of indices

l= {1,...,m} , 2= {m+1,...,2m} and 3: {2m+1,...,3m}.

Let is l, k e 2 be any two indices so that ei, ck are two principal directions
corresponding to the curvatures - «I3 , 0 respectively . Then from (3.2.34) using (3.2.33)

we obtain
(3.2.37) 1: 1+ 1,).,, = - 2m{(c,)m}(ek) — Egaflcixofiej) + gaflekxoflej) .
je3 je )6
From Codazzi equation (3.2.32) we get
J3 df(ei) = -\/3 co';(ej) , 243m}(ek) = 43 m;(ej) , so that

(3.2.33) toga.) = dag.) . ' )=%w‘,‘(e)

Now in (3.2.37) we express everything in terms of (o'i‘(ej) using (3.2.38) and simplifying

to get

(3239) 2 [co'i‘(ej )]2= , for every ie 1, ke 2
je3

By a similar computation, using expressions for 1 + Kill, , where i e 2, k e 3 and,

respectively, is l, k e 3 , we obtain

(3.2.40) 2 [(011%)]2 = 1 , for every ie 2, ke 3
jel

(3.2.41) Z[m‘;(ej2=)] ~31- , forevery ie 1, Ice 3 .

81

Next, we compute tr (VeiA)2 from (3.2.36) to get fori e 1:

tr (VciA)2 = 3 2% [m‘i‘(ej)]2 + 12 2;[aw§(ej)]2
lee E2

= 3m+125§ = 6m.

Similar computation can be carried out for i e 2 and i e 3 , yielding the same result, so

(3.2.42) u(VciA)2= 6m , forevery i=1,...,3m .

Therefore, we see that (3.2.35) is satisfied with p = - 3 and q = 6m . We conclude that all
minimal isoparametric spherical hypersurfaces with three curvatures are of 3 - type via 31' .

As a matter of fact we can show that

3~ 2~ ~ ~ I _
A x + aA x + be +c(x-3m+2) — 0

 

is satisfied for a = — (10 + 24m) , b = 4 [(3m +1)(15m + 6) — 2] and
= - 48m (3m +1)(3m + 2) , so that the three eigenvalues of the Laplacian arising from
the decomposition a: = 2'0 + §p+ 52"! + ii, are 1p = 6m , lq = 2(3m + 1) and
A, = 4 (3m + 2) . 0
Remark. From the above we know three eigenvalues of any isoparametric
spherical hypersurface with three curvatures. Even though the spectrum of the Cartan
hypersurface is known ( [M-O-U] ), not much information is available about eigenvalues of
the Laplacian for other isoparametric hypersurfaces with three curvatures. Also, it is known
that any minimal spherical hypersurface with tr A2 = const has n, tr A2 and n + tr A2 as

three eigenvalues of the Laplacian (cf. [M-O-U] ). See also [Ko], [Mu] .

 

82

In order to chech which minimal isoparametrtic spherical hypersurfaces (or at least
homogeneous ones) with four or six principal curvatures are mass - symmetric and of 3 -
type via 52' , one has to check the condition (3) of Theorem 3.2.1 . That can be done (for
homogeneous ones) by the methods of [T-Ta], considering the action of the Lie group
K = Sp(2) or 02 on the Euclidean space m arising from the Cartan decomposition
g = k + m of the corresponding orthogonal symmetric Lie algebra (g, k, 0), but the
computations involved are rather long. First, one has to choose a point P e a ( a 2 -
dimensional abelian subspace of m ) so that the orbit of P under the adjoint action of K is
minimal in sphere. That requires some manipulation with the roots of the Lie algebra
determined by a . Second, one needs to find the principal directions for the shape operator
and compute the connection coefficients. The shape operator of an orbit hypersurface is
given by AX = - [Y, é] , where § is the unit normal to the hypersurface in sphere
( é is perpendicular to P , and § and P span a ) , and Y e k is a vector such that
x = Y; = [Y, P] (cf. [T-Ta]) .

Also, it would be important to resolve if any minimal spherical hypersurface which
is of 3 - type and mass - symmetric via '1? is necessarily isoparameuic. Techniques used in
this chapter can be modified to study hypersurfaces of a projective space which are of low

type via the first standard embedding of a projective space.

CHAPTER 4

SUBMANIFOLDS OF Em WITH HARMONIC
MEAN CURVATURE VECTOR

In this chapter we discuss certain aspects of the following problem proposed by
B.Y. Chen [C 7] .

Problem: Classify or characterize submanifolds x : Mn —9 Em which satisfy

(4.1) AH=0 .

where, as usual, H denotes the mean curvature thor of the immersion and A, the
Laplacian of M acting on sm00th functions, nattu'ally extended to act on E1“ - valued maps .
Obviously, every minimal submanifold in Em satisfies AH = O , so the real
problem is if there are other submanifolds, besides minimal, that satisfy this equation. In
view of the formula A x = - n H, the equation (4.1) becomes A77t = 0 , that is, we want to
study immersions which are biharmonic (but not harmonic).
The well known theorem of Takahashi (Theorem 1.5.1) asserts that if Ax = Xx then
M is minimal in Em if 2. = 0 , or M is minimal in hypersphere of Em centered at the origin, if
7L > O ( 2.. < 0 cannot occur here ). An analogous problem to this would be to consider the
equation AH = 1H and see what it implies for submanifold M. In particular, for 3. == 0 we
have the problem above. If M is compact , AH = AH implies A(Ax - Xx) = O , so we get
Ax — 7.x = c = const . Further, if A = O , by integrating we have c = 0 , and therefore

Ax = O , which means that the immersion M is minimal. But it is well known that there are

83

 

 

84

no compact minimal submanifolds of E"1 . In case 2. at O, we get A( x + c/A.) = M x + c/A),
so submanifold is minimal in hypersphere centered at - c/X ( i.e. it is of 1 - type) . It is
easy to see, using induction, that condition Al‘H = O (k nonnegative integer) is possible
only on a noncompact manifold (cf. [C 4], Corollary 8.7.2., p. 302 ), and that is what
makes our problem difficult since analysis on noncompact manifolds is not so well
understood. While constructing examples (if they exist ) of nonminimal submanifolds
which satisfy AH = 0 seems to be reasonably difficult , we prove that under various
additional conditions on the immersion x , a submanifold satisfying (4.1) is necessarily
minimal. Let us note that there are known examples of submanifolds in pseudo - Euclidean
spaces satisfying AH = 0 [Ho] . In fact, Houh gave characterization of spacelike surfaces
in pseudo sphere satisfying (4.1) in terms of Weingarten maps.

First we consider a curve case ( n = 1).

Theorem 4.1. If x : C -> Em is a curve with mean curvature vector H satisfying
AH = 0 , then the curve is a straight line, i.e. totally geodesic in Em .

Proof. Let s be a natural parameter of the curve . Then the Laplacian becomes

4
A = - d2/ds2 , and we have 0 = AH = - A7x = - Ell-8%- . Hence, x has to be cubic

polynomial in s, x = %a s3 + %b s2 + c s + d , where a, b, c, d are constant vectors.
Since s is the natural parameter we have

dx dx

ma?

1=<

<a52+bs+c.asz+bs+c >

= lalzs4+2<a,b>s3 + (2<a,c>+lbl2)s2+2<b,c>s+|c|2.

Onthe1ighthand side wehaveapolynomialins,sowemusthave a=b=0, |c|2==1.

In other words, x(s) = cs + d with l c I2 = 1, and therefore the curve is a straight line .0

 

 

85

From now on we assume that the dimension n 2 2 . We use fundamental formula

(2.25) of B.Y. Chen

(2.25) AH: ADH + n An+1 IIZH + a (H) +tr(\"'/AH) ,

where emlll H, a(H) = EMAHAIk, , and U(VAH) = U(VAH) + u(ADH).

r=n+2
{e1, , en, en“, , em} will denote an adapted frame with the usual range for indices.

The mean cm'vature on is defined by H = on en“ . Now we proceed with the computation of
ADH and tr (WtH ) .

n
D
A H = 2 ( DVeiciH ' DciDciH )
i=1
n
= z [ Dvcici(aen+1) — DCiDCi(acn+l) ]
i=1
n
i=1
so that

< ADH , 611+] > = A“ - 2 < DCiDCicfH‘l 9 en+1>
i

i

Aa + all Dem“2 .

Also, we have

u(VAH) = u(VAH) + u(ADH)

 

86

n
z [(VeiAHki + Ancinci]
i=1

2 L [(Vei(°‘An+1))°1 "' ADC.(aen+1)ei 1
i=1 ‘
(4.3)

n

2 I 21c.a)(A,,..e,> + a (Ve,A..1>c. l + a “no...

i=1

n
2A11 +1(Von) + a "ADeml + a 2(V61‘63H1ki
i=1

naVa + 2atrADen+l+ 2An+l(Va) ,

by virtue of the Codazzi equation

Namely, for any X 6 TM we have (see also [C 6])

2 <(VeiAn,1)e,. X > = 2 < ei, (Vcipwpg >
‘ i

tr (VXAml) + z < ei, ADeicmlx >

<nVa + trADch,X>

Assume now that A H = 0 on a manifold. Then by separating off tangential part , normal

part in the direction of °n+1 and normal part perpendicular to em] we get respectively

87
U(VAH)=0 ,
<ADH,en+1> +01 IIAM1112=0 ,
u(AHAr) + <ADH,e,>= o, r=n+2,...,m ,

or, due to the calculations above we see that the condition AH = 0 is equivalent to the

following system of equations

(4.5) 2An+1(Va)+ naVa + 20ttrADen+1 = O ,
(4.6) A01 + alchn+1||2 + a "Am, 112 = o ,
(4.7) a tr(An+lAr) + a < Aben+1 ,e, > — 2 < omen+1 , e, > = o , r = n+2, m.

Because of the equation (4.6) we readily obtain the following lemma .

Lemma 4.1. Let x : M“ —-) E"1 be an isometric immersion and assume that the
mean curvature Otis constant . Than if AH = 0 it follows that at = 0 , that is the
submanifold is minimal.

System (4.5) - (4.7) in general is difficult system of PDE's to solve, but if

D6,,“ = O , in particular if M is hypersurface, the system simplifies to (last equation is not

present in hypersurface case )

(4.8) An+1(V0t) = —gaVa
(4.9) Act + a 11A,,1112: o
(4.10) au(An+lA,) = o , r=n+2, .m.,.

From the equation (4.8) it follows that on the open (possibly empty) set {Va at O } of M,
Va is a principal direction of ¢n+1 and - g a corresponding principal curvature .

88

Theorem 4.2. Let x : M“ —) En+1 be a hypersurface of En+1 with at most two
distinct principal curvatures. Then the condition AH = 0 implies H = O , that is M is a

minimal submanifold of 13"+1 .
Proof. For a hypersurface the condition AH = 0 is equivalent to the system

(4.11) A0: + allAll2= o ,

(4.12) A(Va) = —BaVa .

Let us also recall the Codazzi equations in the form
(1.1.29) (Aj - A,)co}(e,) = ejli , 1;: j
(1.1.30) (11- - 1k) m§(e,) = (7.,- kafiej) , i¢j¢k¢i .

LetUbeanopen set ofMdefined by U= {p6 MlVa2¢O atp},and1et {ei},i=1,...,n

be the basis of principal directions on U so that e1 =ilg—gli is the eigenvector of the shape

operator corresponding to eigenvalue 11= - 1got . Then eja = O for j 2 2 . If the

multiplicity of 1.1 is at least 2, i.e. if Xi = 11 for some i 2 2 then ela = 0 . That follows
from the equation ( 1.1.29) putting j = 1 . In that case a = const , and by Lemma 4.1 we
conclude that a = O on U and as well on entire manifold M. If the multiplicity of 2.1 is one,

then since there are at most two distinct principal curvatures and since tr A = nor we have

3n0t
(4.13) M = 12““ ’ 12 = *3 =-" = A. = 2(n-1)

 

In the rest of the proof, all computations will be done on the set U .

n+8 n20t2 . ,
chompute IIA "2:qu = (7-3173—— . Srnce 21¢ 3.,- and eja = o for] 2 2,

we get from (1.1.29)

89

(4.14) (oii(e1) = o , for all j=1,...,n , i.e. Vele1= o ,

which means that the integral curves of e1 are geodesics on U. For j = 1 and i 2 2 (1.1.29)

gives ( 2.1 - 1i) coli(ei) = elki , with 2.1 - Xi = - Egg-7212? . Therefore,

(4.15) m{(e,) = 311‘— = "3931 , fori22 .
2.1-1i (n+2)0t

For j,k22 ,jaek andi=1wehave xj-xk=o at 3.1-1}, sotheformula (1.1.30)

yields
(4.16) df(cj) = o , for j,k22 ,jaek

Combining (4.14), (4.15) and (4.16) we get the expression for (011‘ as follows

(4.17) (of = "3119—0)“, for 1:22.
(n+2)a

Let us compute now the Laplacian of the mean curvature

Aa = Z [(Veici)°‘ - eieia]

2 [ z (”Reficka‘eicia]
i k

= [2001031)] 61‘ll ’ 6161“

301-12 (61002- 6161a . by (4.15) .

(n+2)a

Therefore, (4.11) becomes

9O

2 3
3911) (61002 __ 6161“ + (n+8)n a = o

(n +2)or. 4 (11-1) ’ or
.. _ 3(n-1) . 2 _ (n-1-8)n2 3 __
(4.18) a (n+2)a (a) 4(n-l) a _ 0 ,

where ‘ denotes derivative with respect to e1 . Formula (4.17) can be rewritten as

(4-17) (n+2)0t(1): = 3a'wk, for k.>_2 .

Differentiating this relation we get

(4.19) (n+2) do A a): + (n+2) a dart = 3 da'A a)" + 3 a' dot)k .

Using Cartan structural equations (1.1.20), (1.1.21) and (1.1.22) we have

d0): = (OMIAmMI + Zcoj

j=l

7.1)., ark/(m + Z 2 m{(e)mg(e,)dxco‘

j=2 r=l

 

311%2 1 _ 3a' m,
" — 4(n-l) (0A0; (n+2)a f: g? (01.03:) Am ’

and

cokAw1+ gag/\m‘

 

=(n+2)a

Because of these calculations, the left hand side of (4.19) takes the following form

 

30t'2 31120:2 _ 3a' 02' (0’
Tm13Awk+(n+2)a[—4—(——-n_1)m/\mk (n+2)“ ggmkm) A ]

and the right hand side of (4.19) reads as

91

9au2

(n+2)0t

 

n
.' 1 ' a n a .
3a 0) A(Ok-l- 3 EZLeJ-(aMt'Atok— (DI/\(Dk-l- 3a' grog/wok
J= i=2

Now comparing the coefficients of (01 A (1)" term (k 2 2) of the left and right hand side of

(4.19) we obtain

 

afi_w_3a. 942 th.
0: 401-1) _ (n+2)0t ’ ans
.. _ MS .2 (1142)::2 3 _
(420) a (n+2)a a + 4(n-l) a - 0

Eliminating a" = elela from (4.18) and (420) we 86‘

2
(4.21) 2%) a'2 + ($11“)— a4 = 0

Clearly, if n 2 4 from (4.21) we conclude a = 0 on U which is a contradiction unless U

is empty . In any case we can solve the equations (4.18) and (4.20) explicitly. Namely, for

any equation of the form y" = f(y, y') , we introduce the substitution v = y' , regarding v
as a function of y . Then y" = v d; , and the equation becomes the first order differential

equation in v(y) . Regarding the equation (4.18) , let 2 = (a')2. Then from the above we
have

_ = 2a'— = 2a'g- = 2a" .

and the equation (4.18) becomes

5.1.2. _ M z _ m a = 0
doc (n+2)a 2(n-1) ’

which is first order linear differential equation whose solution is given by

92

2 m
(4.22) z = (1'2: Wa4+ catn+2 , c=const .
Using the same method for solving (4.20) we get
203.15) 2 2
(4.23) (1'2 = km“2 — lit-'12);- a4 , k = const .

4(n-l)

Considering all possibilities for exponents of a in (4.22) and (4.23) we see that these two

equations contradict each other on the set U, and therefore U must be empty, which means

that the mean curvature at is constant and hence by Lemma 4.1 submanifold M is minimal.

Corollary 4.1. Any surface in E3 which satisfies AH = 0 is minimal .

Corollary 4.2. If Mn is a quasiumbilical hyperesurface of BMI which satisfies
AH = 0 , then M is necessarily minimal and therefore generalized catenoid (see [Bl]) .

By a result of Cartan [c 1], a hypersurface Mn c E"+1 is quasiumbilical if and only if it is
conformally flat (for n 2 4 ), so the Corollary 4.2 can be appropriately stated for

conformally flat hypersurfaces ( except when n = 3 ) .

Next we have following theorem for pseudoumbilical submanifolds

Theorem 4.3. Let x : Mn -—) Em be a pseudoumbilical submanifold , that is

An+1= aI.If AH=0 andn¢4 then MisminimalinEm .

Proof. From the equation (4.5) and pseudoumbilicity we obtain

(4.24) E-2”:ZV01 +“Aoem = O , for onto ,

or, equivalently,

93

(4 25) 912— n m r r
. 2 (eia) + 2 Z con+1(ek) 11k, = o , for every i= 1,2, ...,n .
k=l r=n+2

Using the Codazzi equation (1.1.24) we have

(4.26) (Vcihxcj, ej) = (chhXei, cj) , i¢ j .
We fix index is {1, , n} and let rdenote any index 2 2 . By comparing terms in the
direction of en“ on both sides of (4.26) we obtain
1 . .
Dei( h“; e... + h},- e.) = Delete.) — cage) hng‘ e... — w}(e,->h“;;‘e..1 .

Note that hnfil = 0t for any j , and hni’Jfl = 0 for i¢ j . Therefore, en“ components give

(4.27) (eia) + Zhj’j m";1(e,) = 2h;- co“:‘(c,-) . i¢j
r l'

n
Summingoverall j¢i , andobservingthat0=trA,= 2th , thatis
j=l

n

2 ht. = - 11;, ,we get from (4.27)

.. .11
yr
(n-1)(e,a) - 211;, m“:1(ei) = 2211;,co“;l(ej) , that is
r r jati
(4.28) (n- 1) (eia) + Z 2 mn;,(ek) 11;, = o .

k=1 r=n+2

Comparing (4.25) and (4.28) we see that if n at 4 , eia = 0 for every i = 1, , n

which shows that or = const and therefore equal to zero. 9

Lemma 4.2. Let x : M“ -) Em be an isometrically immersed submanifold which

satisfies AH = 0 and < x, H > = const . Then M is minimal in E“.

 

 

94

Proof. By formula (1.5.4) we have

A<x,H>= -n<H,H> +<x,AH>+2trAH = —na2+2na2= naz.

Therefore,if <x,H>= const ,weget 0t=0 .

As a corollary, any cone in Em that satisfies AH = 0 must be minimal . Namely, without

loss of generality, we can assume that a cone has the vertex at the origin so that

< x, H > = const holds .

Next we show that if AH = 0 for a submanifold M c Em and M is of finite type, then

H = 0 , so M is minimal again. First, using induction we can easily prove the following

Lemma 4.3. If M is a manifold and A the Laplacian acting on smooth functions of
M . then no eigenfunction ( not identically 0) of A can be represented as the sum of k

( k 2 2) other eigenfunctions from k different eigenspaces.

Theorem 4.4. Suppose that A’H = 0 holds for a submanifold x : Mn -> Em ,

for some positive integer r. If M is of finite type it follows that M is minimal , i.e. of null
1 - type .
Proof. Suppose that M is of k - type so that we have

(4.29) x = x0+ xtl+ + xtk .
with x0: const and A xti= Mix. , i 2 1 .Then taking A“1 of (4.29) we have
_ _ r = r+1 ____ r+1 r+l
0— nAH A x 1,1 xt1+...+7sthtk.

By Lemma 4.3 , this is possible only when there is only one nonzero x,i in this sum , and
the corresponding eigenvalue 2.“ is zero . This means Ax = 0 and the submanifold M is of
null 1 - type (minimal) . O

 

 

95

Corollary 4.3. Suppose that x : Mn —9 E1“ is a submanifold such that the
component functions of x are eigenfunctions of the Lapacian. If AH = 0 then M is
minimal.

Proof. If x = (x1, , xm) and Axi = Mi xi then x is of finite type, actually of
type S m since

x= (x1,0,...,0) + (0,x2,...,0) + (0,...,0,x ) .

m

Then the Theorem 4.4 proves the claim. 9

Surfaces of revolution in E3 which have the property [described in previous

corollary were studied in [G] .

In view of Theorem 4.2 it seems probable that a hypersurface of E‘"[1 which satisfies
AH = 0 is minimal since there is no "room" in the normal space. (There is also strong
indication that that is so for any 3 - dimensional hypersurface in E4.) If a codimension is

higher, it is possible to have a nonminimal submanifold which satisfies (4.1), but

construction of such submanifolds seems to be difficult. If H = (hl, , hm) and AH = 0,
then each hi is harmonic. For a harmonic map on a manifold there are Liouville type
theorems. For example, if M is a complete Riemannian manifold of nonnegative Ricci
curvature, then any bounded harmonic function on M is a constant function. The same

conclusion holds if a harmonic map grows slower than linear growth or have a finite
energy (see [Y], [Che]) . So, if such submanifold satisfies (4.1) and the mean curvature at

is bounded then a = 0 . Also a bounded harmonic map on a simple Riemannian manifold is

necessarily constant (see [Hi]) . A manifold M is simple if it is topologically Rn with

metric for which A is uniformly elliptic.

 

 

 

SUMMARY

For an isometric immersion x : M“ —) F.m of a Riemannian manifold into a

Euclidean space, one defines the map if = x-xt ( x regarded as column vector) from M into
SM(m), the set of m x m symmetric matrices, which we call quadric representation of M.
If M is submanifold of the unit hypersphere centered at the origin (henceforth called
spherical), then it is well known that r is an isometric immersion (via 2nd standard
immersion of sphere). It appears, however, that this map has not been studied in general.

A smooth map f : Mn -) R is said to be of k—type if it can be decomposed as
f = f0 + if[ (k nonzero terms in the sum) , where f0 = const and Aft: 2t, f,i.e. ft's
are eigenfunctions of Laplacian on M. This naturally extends to an Em-valued map. In
particular, a manifold immersed into Euclidean space is of k-type if the corresponding
immersion is so.

In Chapter 2 we proved some general results about quadric representation. First we
showed that 'x' is an isometric immersion if and only if M is spherical. The same conclusion
if ‘x' is conformal (n 2 2) (see Theorems 2.1 - 2). Submanifolds for which 2' is l-type map
are classified as totally geodesic spherical submanifolds (Theorem 2.3) . While it is
relatively easy to construct nonspherical submanifolds for which if is finite type (Example
2.1), we prove that for minimal submanifold of Em the quadric representation is of infinite
type (Theorem 2.4 ). For a spherical submanifold, certain relationships between the
immersions x and 'x' can be shown as exemplified by

Theorem 25. Let x: M“ -) 3""1 c: Em be an isometric immersion and let

96

 

 

97

'x' : Mn -) SM(m) be its quadric representation. Symbols with ~ are related to the
immersion 'x' , those without ~ to the immersion into E"1 and symbols with ' relate to
the immersion into 8""1 . Then
i) llfill=const a Ilhll=const,

ll H II =const c: IIH ll =const
ii) Mn is pseudoumbilical in SM(m) via 32' a M“ is pseudoumbilical in Em via x.
iii) 5H = 0 c: h'=0 , i.e. Mn is totally geodesic in 8“"1 .
iv) VS =0 es h'=0.

In Chapter 3 we study compact spherical hypersurfaces which are of low type via
the quadric representation. We have the following classification result for those which are

of 2 - type via 31' , thus generalizing similar result of M. Barros and B.Y. Chen [B-C].

Theorem 3.1.1. Let x : Mn -) Sn+1 be an isometric immersion of a compact
n - dimensional Riemannian manifold M into Sn+1 (n 2 2) . Then i' = xxt is of 2 - type
if and only if either
(1) M is a small hypersphere of S“1 of radius r < 1 , or
(2) M = Sp(rl) x Sn'p(r2) , with the following possibilities for the radii r1 and r2 :

Ur? ELI 2-2:Pil.u)121_2f_2_ 22:2

n+2’r2"n+2’11 l— 2 12sz

" n+2 ’ 1p2=n+2 ; iii) r} = n+2 ’ r2: n+2
The immersions in (1) and (2) are given in a natural way .

Next we compute the third iterated Laplacian and undertake study of minimal

spherical hypersurfaces which are mass - symmetric and of 3 - type via ‘1? . We obtain the

following characterization

 

T":—

98

 

Theorem 3.2.1 Let x : M“ -> S“1 be an isometric immersion of a compact
manifold M“ as a minimal hypersurface of Sn+1 . If i' is mass - symmetric and of 3 - type
then
(1) trA=trA3=o,

(2) tr A2 and tr A4 are constant ,
(3) u(VxA)2 = <A2X,A2X>+ p< AX, AX>+ q<X,X>, Xe TM
where p and q are constants (depending on the order of M , tr A2 and tr A4) .

Conversely, if (1), (2) and (3) hold then M is mass - symmetric and 1 - , 2 - , or 3 - type

~

viax.

The main result of Chapter 3 is the classification of compact minimal spherical
hypersurfaces which are of 3-type and mass - symmetric via r in dimensions 11 S 5 , thus
giving a new characterization of the Cartan hypersurface SO(3)/Z2 x 22 in terms of its

Spectral behavior. Namely,

Theorem 3.2.2. Let x : Mn -) Sn+1 be compact minimal hypersurface of S“1 of
dimension 2 S n S 5 . Then a: is mass - symmetric and of 3 - type if and only if n = 3 and
M3 = 30(3)”,2 x 2.2 is the Cartan hypersurface .

Actually, all minimal isoparametric spherical hypersurfaces with three distinct

principal curvatures are of 3-type and mass-symmetric via 32' (Lemma 3.2.3).

In Chapter 4 we study submanifolds x : M“ -) E‘m of a Euclidean space with
harmonic mean curvature vector , i.e. those that satisfy AH = 0 , or equivalently Azx = 0 .
Minimal submanifolds being the trivial solution, the real problem is to find nonminimal

examples, that is, those immersions which are biharmonic but not harmonic. While the

99

construction of such examples (if they exist) seems difficult, we show that submanifolds
satisfying AH = 0 are necessarily minimal if any of the following conditions is satisfied

(1) Mn has constant mean curvature .

(2) Mn is a hypersurface of 13““ with at most two distinct principal curvatures .

(3) M“ is conformally flat hypersurface of BMI (n at 3) .

(3) Mn is a pseudoumbilical submanifold of Em ( n at 4) .

(4) M'1 is of finite type .

 

 

BIBLIOGRAPHY

 

[A]

[B-C]

[B-C-G]

[B-U]

[B-G-M]

[B]

[B6]
[B1]

[Car 1]

[Car 2]

[Car 3]

[Car 4]

[Car 5]

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[Ch-E]

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