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LIBRARY
Michigan State
University

 

 

 

This is to certify that the

dissertation entitled

Submanifoids of Euclidean Spaces with Pianar
or Heiical Geodesics Through a Point

presented by

Young Ho Kim

has been accepted towards fulfillment
of the requirements for

W .- J" .
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be charged if book is

 

 

 

returned after the date
stamped below.

 

,,_.__-._ -

 

 

SUBMANIFOLDS OF EUCLIDEAN SPACES WITH PLAN AR OR HELICAL
GEODESICS THROUGH A POINT

By

Young Ho Kim

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1988

 

50 :9- 9,9.3/

ABSTRACT

SURFACES OF EUCLIDEAN SPACES WITH PLANAR OR HELICAL
GEODESICS TROUGH A POINT

By

Young Ho Kim

Compact connected surfaces in 3-dimensional Euclidean space E3 with helical

geodesics through a point are characterized as standard spheres. If the ambient manifold is

» a 4-dimensional Euclidean space E4, then compact connected surfaces with helical

geodesics through a point are characterized as standard spheres which lie in E3 or pointed
Blaschke surfaces which fully lie in E4 . Such surfaces in 5-dimensiona1 Euclidean space
E5, are characterized as standard spheres lying in E3 or pointed Blaschke surfaces which lie
fully in E4 or E5. We also prove that compact connected surfaces in a Euclidean space Em
(m 2 5) with helical planar geodesics through a point must lie in E5 and these surfaces are
one of model spaces above.

A surface in E3 is locally a surface of revolution if and only if it has a point
through which every geodesic is a normal section. So a complete connected surface in E3
is a surface of revolution if and only if there is a point of the surface through which every
geodesc is a normal section.

If surfaces in Euclidean space have planar geodesics through a point, then we
show that those geodesics are normal sections and the Frenet curvatures of them are
independent of the choice of the direction. Furthermore, a neighborhood of the point is
completely determined by the Frenet curvature of a fixed geodesic.

Finally, we define a normal section of a pseudo-Riemannian submanifold in
pseudo-Euclidean space in a way similar to that of the Riemannian case. Surafces in
pseudo-Euclidean space with planar normal sections are classified as Veronese surfaces or

flat surfaces if the surfaces do not lie in a 3-dimensional affine space.

ACKOWLEDGMENTS

I wish to express my deep gratitude to my advisor, Professor Bang-Yen Chen. All
of my work has been done by his patient guidance and many valuable suggestions. 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 in my graduate
studies. Finally, I especially thank my wife, Hieyeon, and my parents for their

unwavering love and encouragement.

INTRODUCTION
CHAPTER 0.
CHAPTER 1.

CHAPTER 2.

CHAPTER 3.

CHAPTER 4.

SUMMARY
BIBLIOGRAPHY

TABLE OF CONTENTS

Preliminaries

Surfaces of a Euclidean space with helical geodesics
through a point

Characterization of surfaces of revolution in a three-
dimensional Euclidean space

Surfaces of a Euclidean space with planar geodesics
through a point

Surfaces in a pseudo-Euclidean space E? with

planar normal sections

iii

21

55

62

79

101
104

INTRODUCTION

The study of surfaces in a Euclidean space is fundamental to the understanding of
geometric models and provides good ideas from which to develop certain theories. So,
many geometers tried to study surfaces in Euclidean space from many different points of
view. The theory of pseudo-Riemannian manifolds enables us to study some surfaces in a
pseudo—Euclidean space.

In this sense, surfaces in a Euclidean space or a pseudo-Euclidean space with some
properties concerning geodesics and normal sections need to be examined and classified.

Helical submanifolds were first introduced in [ Be.A ]. Helical submanifolds in a
Euclidean space or a unit sphere have been studied by K. Sakamoto, [ S-l ], [ S-S ], [ S-6
], since 1982. He proved that such submanifolds are either Blaschke manifolds or
Euclidean planes.

On the other hand, in 1981, B.-Y. Chen and P. Verheyen, [ Ch.B-V—l ], [ Ch.B-
V-2 ], introduced the notion of submanifolds with geodesic normal sections and classified
surfaces with geodesic normal sections in a Euclidean space. They also proved that helical
submanifolds have geodesic normal sections. Later, P. Verheyen, [ V ], proved that a
submanifold M in a Euclidean space Em of dimension m with geodesic normal sections are
helical. So the concept of submanifolds with geodesic normal sections coincides with the
concept of helical submanifold if the ambient space is a Euclidean space.

In [ Ho.S ], S. L. Hong introduced the notion of planar geodesic immersions.
Such irnmersions were later classified by J. A. Little, [ L], and K. Sakamoto, [ S-l ],
independently, who proved that Mn is a compact symmetric space of rank one and the

second fundamental form is parallel. The Veronese surface can be considered as one of the
1

best examples determined by the planar geodesic immersion if the ambient space is a 5-
dimensional Euclidean space E5.

Using the theory of pseudo-Riemannian structures, C. Blomstrom, [B-2 ], defined
planar geodesic immersions of pseudo-Riemannian submanifolds into a pseudo-
Riemannian manifold and she classified complete parallel surfaces with planar geodesics
in a pseudo-Euclidean space E? as pseudo-Riemannian spheres, flat quadric surfaces or
Veronese surfaces.

However, there has been no research on a submanifold M in a Euclidean space Em
with the property that for a fixed point p in M every geodesic passing through p is a helix
of the same curvatures or every geodesic through p is planar or a normal section at the point
p. From this point of view, we are going to study surfaces in a Euclidean space which
have such properties and to chracterize such surfaces. Furthermore, we are going to
classify surfaces of a pseudo-Euclidean space with planar normal sections.

In Chapter 0, we introduce some fundamental definitions and concepts which are
the necessary background for the theory throughout this thesis.

In Chapter 1, we study compact connected surfaces in a Euclidean space with
helical geodesics through a point . If the ambient space is a 3-dimensional Euclidean space
E3, then such surfaces are characterized as standard spheres. If the ambient manifold is a
4-dimensional Euclidean space E4, then we obtain that geodesics through the point must be
of rank 2, i.e., they are planar curves, and surfaces are characterized as standard spheres
which lie in E3 or pointed Blaschke surfaces which fully lie in E4. If the ambient manifold
is a 5-dimensional Euclidean space E5, then geodesics of the given surface through the
point may be of rank 4. In this case, using some fundamental equations obtained from the
helices through the point, we set up a system of ordinary differential equations. By solving
this system of differential equations, we have examples of pointed Blaschke surfaces

which are diffeomorphic to a real projective space and lies fully in E5. So we can

characterize such surfaces as standard spheres which lie in E3 or pointed Blaschke surfaces
which lie fully in E4 or E5. By means of this characterization, we have a new
characterization of the Veronese surface, namely, a Veronese surface is characterized as a
compact connected surface with constant Gaussian curvature and a nonumbilical point
through which every geodesic is a helix .

In Chapter 2, we study a surface M in a three-dimensional Euclidean space E3 with
geodesic normal sections at a point p. By adopting geodesic polar coordinates about the
base point p, geodesics are proved to depend only on the arc length and thus M is
characterized as locally a surface of revolution around the point p. So, if M is complete
connected, then M is a surface of revolution if and only if there is a point p through which
every geodesic is a normal section.

In Chapter 3, we study a surface M in a Euclidean space Em with planar geodesics
through a point p. We prove that planar geodesics through a point p are normal sections of
M at p. We also prove that geodesics through p only depends on the arc length and thus
Frenet curvatures are independent of the choice of the direction. So, we can precisely
determine how the surface looks like in a neighborhood by means of the Frenet curvature
of a fixed geodesic through p. We also observe that a surface in a Euclidean space Em with
planar geodesic through p is possible to lie fully in a considerably higher dimensional
Euclidean space En c: Em if p is an isolated flat point .

In Chapter 4, we define a normal section of a pseudo-Riemannian submanifold in
a pseudo-Euclidean space in a way similar to that of the Riemannian case. We also define
pseudo-isotropy which is a similar notion to isotropy in a Riemannian manifold and obtain
that if a surface M in a pseudo-Euclidean space E? with planar normal sections is not
contained in a 3-dimensional affine space, then all normal sections are geodesics and hence

M has planar geodesics. Using the classification theorem of C. Blomstrom, [ B-2 ], we can
classify surfaces of index r in a pseudo-Euclidean space E? with planar normal sections as

4
. . . 2+1: . 5
flat surfaces Wthh fully he in Elk (k = 2, 3) or Veronese surfaces 1n E “34) or

E; r +1)(2_ r) if the surfaces are not contained in a 3—dimensional affine space.

CHAPTER 0
PRELIMINARIES

Let M be a smooth manifold and let c°°(M) be the set of all smooth functions
defined on M. Then C°°(M) becomes a R-module, where R is the real number field.

Definition. A tangent vector to M at a point p is a function Xp : C°° (M) —> R,
whose value at {e c°°(M) is denoted by xp(f), such that for all r, g e c°°(M) and

r e R,
(i) Xp(f + g) = Xp(f) + Xp(g).
(ii) Xp(rf) = r Xp(f)
and
(iii) Xp(fg) = Xp(f) g(p) + f(p) Xp(g).

We usually regard Xp(f) as the directional derivative of f in the direction Xp at p.

The set of all tangent vectors to M at p is called the W to M at p, which is
denoted by Tp(M). The tangent space Tp(M) defines a vector space of dimension n over R.
A (smooth) vector field on M is a C°° mapping which assigns to each point p of M a
tangent vector to M at p. The set of all tangent vectors on a manifold M of dimension n is
called the tangent bundle, denoted by TM, which forms a fibre bundle over M with Tp(M)

as the fibre over a point p in M.

6

A manifold M on which there is a symmetric tensor field g of type (0,2) which is
positive definite and bilinear is called a Riemannian maeifeld and g the Riemaenieg meeje
or the first fundamental ferm, If there is a nondegenerate bilinear tensor g of type (0,2)
with the property that the dimension of the negative definite subbundle of TM with respect
to g is constant on M, then M is called a Wm a semi-Riemannian
menifeld and g the pseede-Riemennian meme. In this case, the dimension of the negative
definite subbundle is called the i_nd_g<_ of the manifold M. If M is a pseudo-Riemannian
manifold, we will use the notation M? as M that is of dimension n and has index r. A
pseudo—Riemannian manifold of index one is called Lerenteian. The Nlinkowski space-
time Ell1 is the best known example of a Lorentzian manifold and is largely dealt with in the

special relativity. If the index is zero, then M3 = Mn is nothing but a Riemannian

manifold.

The metric tensor on a manifold defines the lengths of vector fields and angles
between them. If the metric is indefinite, then a nonzero vector field may have zero or
negative length. We shall say that a nonzero vector field is spacelike if it has positive
length, dm_eli_ke if it has negative length and fighdilge or mfl if it has length zero. We regard
a zero vector as a space like vector.

We now define a connection on a manifold.

Definition. A eenneetien V on a manifold M is a mapping of the product of the
set of vector fields into the set of vector fields denoted by V(X, Y) = VXY which has the

linearity and derivation properties : For all f,g e C°°(M) and all vector fields X, Y and Z,

(i) V Z = f VXZ + g VYZ,

fX +gY

(ii) VX(fY + gY) = (Xf)Y + N X Y + (Xg)Z + g VXZ,

VXY is called the eevaridnt derivadve of Y with respect to X for the connection V.

Then we can define the so-called tersien T :

T(X,Y) = VX Y - VYX - [X, Y],

where [X, Y] is the vector field defined by [X, Y]f = X(Yt) - Y(Xf) for all fe C°°(M).

It is known that there is a unique connection V on a pseudo-Riemannian manifold

satisfying
(iii) T = 0,
(iv) X<Y,Z>=<VXY,Z>+<Y,VXZ>

for all vector fields X, Y and Z , where < , > is the pseudo-Riemannian metric tensor.
From now on, < , > always means a Riemannian or a pseudo-Riemannian metric tensor
according to the context unless it is stated otherwise. The connection satisfying above (i) ~
(iv) is called the Lexi-inita ednneetjen.

The covariant derivative of a general tensor field S of type (r,s) is naturally defined
by linearity and derivation as follows : Since S can be regarded as a multilinear mapping of

TM x TM >< x TM, s copies of TM, into the space of contravariant tensors of

degree r,

S
(VXS)(X1,~", XS) = VX(S(X19°N9 XS)) ' iEIS(X1,..., VXXI,..., XS)

for any X1,..., XS 6 TM. By setting

(VXS)<X1,..., xs) = (stx, x1.-. X.)

we obtain a tensor field VS of type (r, s+1). We shall say that a tensor field S is parallel if

VS = 0. So the metric tensor field on a pseudo-Riemannian manifold is parallel with

respect to the Levi-Civita connection.

We now define a (regular) curve 7 on a manifold M.

Definition. A curve is a smooth mapping 7: I —> M, where I is an open

(1
interval. In particular, if E:- 49 0 for all t e I, then 7 is said to be rem. In the sequel, a
t

. . . d (1
curve means a regular curve unless 1t rs stated otherwrse. Lets = Ind—III dt, where ”21%,“2

t0

= <%¥-, Egg-z Then sis the arc length defined on the curve 7. If a curve is parametrized by

arc length, then it has the unit speed.

The existence of a Riemannian metric or pseudo-Riemannian metric on a manifold
provides an important tool in the study of manifolds from a geometric point of view,
allowing us to introduce on such spaces many concepts of Euclidean geometry such as
distances, angles between curves, areas, volumes and straight lines. Developing the idea
of a straight line in a Euclidean space, we define a geodesic 'y in M as a curve 7
parametrized by arc length s such that 7 satisfies

dth h dxj dxi

dsz 1” fifties—K =0,

 

where {xh} is a local coordinate system and Va/axi 8/ij = E FE a/axh, y(s) =(xh(s)) .

9

Thus, 7 is a geodesic if and only if VTT = 0, where T = "3% . In other words, a curve 7 is a

geodesic if and only if its velocity vector field is parallel along y.

We now give the definition of the exponential mapping.

Definition. Let p e M and X be a unit vector in Tp(M). Let y(t) be the geodesic
emanating from p with initial velocity X with domain (a,b). Set expth = y(t) for

t e (a, b). expp is called the expenenp'al mapping at p.

exp p carries lines through the origin in Tp(M) to geodesics through p e M. Thus,
the distances in M near p are approximated by distances in Tp(M). As a matter of fact,

expp gives a diffeomorphism from a neighborhood of the origin in Tp(M) onto a

neighborhood of p in M. For a curve 7 in E3 parametrized by arc length 8, the length
Ily” (5)” measures how rapidly the curve pulls away from the tangent line at s in a
neighborhood of s. This measurement K(s) = IIY' (5)" is called the curvature of y at s. At a
point where 1((s) at O, a unit vector N(s) in the direction Y" (s) is well-defined by the
equation 7" (s) = 1((s) N(s). Moreover, 'y" (s) is normal to Y(s). Thus, N(s) is normal to
Y(s) and is called the principal normal vector. If Y' (s) at 0, the number 1: (s) defined by

B’(s) = -t (s)N(s) is called the torsion of 'y and B(s) = T(s) x N(s) the binormal vector at

 

y(s). Then we have the following so-called Frenet formulas:

T'(S) = K(S)N(S),

N'(s) = -K(s)T(s) + 1: (s) B(s),

B'(S) = - ’t (S)N(S).

10

where T(s) = y'(s). Generalizing this idea, we obtain the Frenet frame and Frenet

curvatures for a curve 7: I —> M. Let Y(s) = T1(s) be the unit tangent vector and put K1 =

II VTlTlll. If K1 is identically zero on I, then 7 is said to be of rale. If K1 is not
identically zero, then one can define T2 by VTlTl = 1(sz on I1 = { s e I | Kl(s) at 0 }.
Set K2 = II VTIT2 + KlTlll. If K2 is identically zero on 11’ then 7 is said to be of rank
2. If K2 is not identically zero on 11’ then we define T3 by VTsz = -K1Tl+ K2T3.
Inductively, we can define Td and Kd = II VTle + Kd_le_lll and if Kd = 0 identically on

Id_1 = {s e I | Kd_1(s) ¢ 0 }, then yis said to be ofrank d. If'yis ofrank d, then we have a

matrix equation

VT1(T1 ,T2 rd) = (T1 ,T2 rd ) A,

on I d_ 1, where A is a d x (1 -matrix defined by

(0 «10 \
x1 0-K2 0 O
(0.1) A: K2 0

\0 . K(1.1 0-)

 

 

The matrix A, {T1 , T2 ,..., Td} and K1, ..., Kd are called the Frenet formula, Frenet

frame and Frenet curvatures of 7 respectively.

We now define the curvature tensor R on a pseudo-Riemannian manifold. It is a
fundamental theorem of advanced calculus that the second order partial derivatives are

independent of the order of differentiation:

11
a Bf 3 8f

a? (5:5) ran (are)

for all C2 function f. For functions on manifolds the similar property X(Yf) = Y(Xf) does

not hold in general. As matter of fact [X,Y] measures the extent by which it fails:
X(Yf) - Y(Xf) = [X,Y]f, f e C°°(M) , X, Y 6 TM.

Of course, in general VX(VY Z) = VY (VXZ) is not true. To measure this failure of

interchangeability, we define R by

(0.2) R(X,Y)Z = VXVY Z - VYVX Z - V[X,Y]Z’ X, Y, Z 6 TM.

R is a tensor field of type (1, 3) which is called the curvature tensor. For a surface, the
Gaussian curvature is intrisically defined. Generalizing the Gaussian curvature, we define

the sectional curvature.

Definition. Let M: be a pseudo-Riemannian manifold (0 S r S n ). Suppose p e

M: and let II be a nondegenerate plane spanned by X and Y in Tp M11. The seetienel

curvature of the plane II is given by

 

Kpm)=

Kp(II ) is well-defined and independent of the choice of basis of TI . It is not hard to show

that the Gaussian curvature of a surface in E3 agrees with the sectional curvature. If
Kp(1I ) is independent of the choice of the nondegenerate plane II, then we say that M? has

W142. Let K be a real number. We say that M? has W K

12

if M]; has constant curvature K at every point of M2. The famous F. Schur's Theorem

tells us that if a connected pseudo-Riemannian manifold M: of dimension n 2 3 has

constant curvature at every point, then M? has constant curvature, that is, the sectional

curvature is independent of the choice of the point and nondegenerate plane . Complete

simply connected spaces of constant sectional curvature are called real space forms.

Manifolds M and N are said to be isometric if there is a diffeomorphism (p from M to N that
preserves metric tensors. If two surfaces have the same Gaussian curvature, then they are

locally isometric. In [Wo], any pseudo-Riemannian space form is shown to be isometric to
a pseudo-Euclidean space E21, a pseudo-Riemannian sphere 8:6) or a hyperbolic space

H:(r). These latter two are described as follows : Given 0S s S n with 2 S n and a number

r >0, we define

s n+1
(0.3) 530) = {x e E“? l — 2x,2 + 2x9 :14}
i=1 i=s+l

which is analogous to an ordinary sphere

n+1
Sn(r) = { x e En+1 I 2m2 = r2 } = 83(r).

i=1

S:(r) is called the pseudo-Riemannian sphere. We also define the so-called hyperbolic

space
n _{ n+1 s+1'2 n+1.2_ 2}
(0.4) Hs(r)— xe Es+1|-le + 2x1 —-r .
i=1 i=s+2

. . . n .
In the Rremannran case, we have two copres HO of the hyperbolic spaces Hn.

13

We denote a real space form by M(c). Then M(c) has its curvature tensor of the

form
R(X, Y)Z = c(<Y, Z> X - <Z, X> Y)

for X, Y, Z 6 TM(c). So, the usual Euclidean space and the pseudo-Euclidean space have

curvature tensor R = 0, and the ordinary sphere 811(1) has curvature tensor
R(X, Y)Z = <Y, Z> X - <Z, X> Y
and HT'(1) has

R(X, Y)Z = <Z, X>Y - <Y, Z> X.

All of the preceding geometric concepts depend only on the pseudo-Riemannian or
Riemannian metric and not on any external consideration. However, if we consider a
manifold which is immersed into another, then we can observe the image of the immersion
as viewed from the ambient manifold. Naturally we need some theoretical background for
the submanifold theory.

Let M and M be manifolds of dimension n and m respectively (n < m) and let (p : M
—> M be a differential mapping such that its differential (dcp)(p) = ((p *)P is injective at every
point p in M. Then we say that M is an immersed submanifold of M and (p is an
immersion from M to M . If the pullback (p*(g) of the metric tensor g of M is a metric on
M, then M is called a pseude-Riemannien 5p emanifeld of M . The index of the pseudo-
Riemannian submanifold is at most that of the ambient manifold. By observing the

submanifold and the ambient manifold, we may obtain certain intrinsic properties of the

14

submanifold which come from that of the ambient manifold. Let M be a pseudo-

Riemnannian manifold of M endowed with Levi-Civita connection V . We may identify

the image of M with M itself and hence we no longer distinguish vector fields on M from

the images under the immersion. Let X, Y be in TM. Then we can write the covariant

derivative VXY as
(0.5) VXY = VX Y + o(X, Y),

where VX Y is a vector field tangent to M and 0(X, Y) is a vector field normal to M.

Then V turns out to be the Levi-Civita connection on M and o is a symmetric bilinear form
on TM x TM which is called the second fundamental form of the submanifold. (0.5) is

referred to as the Gauss formula for M in M . Let E be a normal vector field on M and X a

vector field on M. We may then decompose Vxl'; as
(0.6) Vxé = -A§X + Dxét

where -A§X and DXE are the tangential component and the normal component of Vxé

respectively. Ag is called the Weigarten map associated to § and forms a self-adjoint
tangent bundle endomorphism on TM satisfying < AgX, Y> = < 0(X, Y), g > for X,

Y 6 TM. D is a metric connection in the normal bundle TiM of M in M with respect to the
induced metric on TJ'M. D is called the nermel eenneetien on M. The equation (0.6) is
called the Wein garten fermpla.

A submanifold M is said to be W if every geodesic in M is a geodesic
in M . It is well-known that M is totally geodesic if and only if the second fundamental

form 0 vanishes identically.

For a normal section E, on M, if A g is everywhere proportional to the identity

transformation I, that is, Ag = pI for some function p, then g is called an umbilical section

15

on M, or M is said to be umbilical with respect to g. If the submanifold M is umbilical with

respect to every local normal section in M, then M is said to be Wei.
Let X1...., Xn be an orthonormal basis of the tangent space Tp(M) at the point

p6 Mandlet

l
H = H 8i 6(Xi. Xi).

where 8i = sgn <Xi, Xi> = :1. Then H is a normal vector which is called the mean

Wp. Let E1...” imm be an orthonormal basis of the normal space Ti'M at

p e M and let Ax = Agx, then H can be written as

m

H = i 2; (trAx)§x.
i=1

It is easily shown that H is independent of the choice of the orthonormal basis fix.
A submanifold M is called a minimal 5;; emanifeld if the mean curvature vector
vanishes identically. We call the submanifold a psepde-pmpilieel 5g emanifeld if the

Weingarten map AH associated with the mean curvature vector H is proportional to the

identity transformation.
If we use the Gauss formula and compute the curvature tensor R of the ambient
manifold, then we obtain the W

(0.7) <R(X, Y)Z, W> = <1'i(x, Y)Z, W> + <o(X, W), o(Y, Z)>

- <o(Y, W), o(X, Z)>

16

and dreQQtlaza'equau'Qn

(0.8) (R(X. Y)Z>i= (VXox Y. Z) - (Vyoxx, Z)

for all X, Y, Z, W 6 TM, where J' denotes the normal component relative to M and VXG

is the covariant derivative of G on TM 6 TLM defined by

(0.9) (onx Y, Z) = on(Y, Z) - 6(VXY, Z) - 0(Y, VXZ)

for X, Y, Z 6 TM. If the ambient manifold has constant curvature c, then the Gauss and

Codazzi equations are given respectively by
(0.10) <R(X, Y)Z, W> = c(<X, W><Y, Z> - <Y, W><Z, X>)

+ <o(X, W), o(Y, Z)> — <o(Y, W), o(X, Z)>
and.

(0.11) (on)( Y, Z) - (VYexx, Z) = 0.

Making use of the Weingarten formula, we obtain the Rieei egpap'en : For X, Y
6 TM and g, n e T‘J‘M

(0.12) <1'i(x, Y)§, n > = <RN(X, Y)§, n > - <[A§, An] x, Y>,

where RN(X, Y)§ .-. DX DYE, - DYDXt; - D[X,Y]§ and [Ag An] = AgAn - AnAg.

17

From these equations we see that the curvature tensors of the submanifold and the

ambient maniofld are related in terms of the second fundamental form.

From time to time, a lot of geometric properties are explained with a standard
sphere S“, a real projective space RP“, a complex projective space CP", a quatemion
projective space HPn and Cayley projective plane 0 P2 as examples. So we need some

basic concepts and definitions concerning these spaces.

Let U be a neighborhood of the origin in Tp(M) such that expplu is a
diffeomorphism. We define s : U —-> U by s(X) = -X and set 3;, =(expPIU)oSo(expp|U)'l.
This Sp is called the Wwith respect to p on expp(U).

The manifold M is called locally symmetric if for each p e M there exists U, a
neighborhood of the origin in TpM, such that the geodesic symmetry sp is an isometry.

Locally symmetric spaces are charaterized by VR = 0, i.e., the curvature tensor is covariant

constant.

Definition. A connected Riemannian manifold M is a WK for each

p e M there exists an involutive isometry sp : M -) M such that

spoexpp = exppos.

Clearly, if M is symmetric, then M is locally symmetric.

Definition. The rank of a symmetric space is defined as the maximal dimenison

of the flat submanifolds, i.e., the curvature tensor R = 0, which are totally geodesic in M.

S“, RP“, CP“, HPn and 0 P2 are the only examples of compact rank one symmetric

spaces. These are abbreviated by CROSSes.

18

Let M be complete and connected The Hopf-Rinow Theorem tells us any geodesic
segment can be extended indefinitely. In general, a geodesic joining two points is not
unique, for example, on a standard sphere 82 in E3, geodesics are great circles and so the
number of geodesics joining the north pole and the south pole is infinite. However, for
two points which are sufficiently close, the geodesic segment joining these two points is
unique. So we need to define the following cut-map : Let X be an element of the unit
tangent space UpM = {X 6 TM I II X H = 1} and let ybe a geodesic emanating from p with
initial velocity X, i.e., 7(8) = expp(sX), where sis the arc length. Let Seg(p,q) be the set

of all geodesics from p to q which are parametrized by arc length. Then for 3 small
enough, Seg(p=’y(0),'y(s)) contains only one element “[0 s]. The set

A = { s e R+| Was] 6 Seg (P=Y(0).Y(S)) }

is necessarily R4. or an interval (0, r] for some r e R+, where R, is the set of all positive
real numbers. If A = Re, we say that there is no cut-point on y; if A = (O, r], we say that
y(r) is the eut-point of p and r the cut-value of 7. Let UM be the unit tangent bundle over
M. The cut-map ¢ : UM —> R,. u {...} defined by ¢(X) = r ifA = (0, r] and ¢(X) = oo if

A = R+. The cut-map is continuous. (of. [K- N], Vol. H, p. 98 ).

Definition. The epkloeps Cut(p) of a point p in M is the set of all cut—points of p,

i.e,
Cut(p) = 1 expp(¢<X)X) I X e- Up(M> }.

Definition. A Riemannian manifold M is said to have spherieel ept—leeps at p if
for every X e Up(M) the cut-value ¢(X) is finite and does not depend on the choice of X.

19

For two distinct points p and q in M we define the link from p to q to be

A(p, q) = { ii“) e um I ve Seg(p, q>}.

 

A subset ('3 of the unit sphere S of a Euclidean space V is said to be a W
if there exists a subspace W of V such that ('3 = S n W. By definition, the dimension of
("3 is dim W - 1.

Definition. A compact Riemnannian manifold M is said to be a Blaschke
manifeld at the peint p in M if for every q in Cut(p) the link A(p, q) is a great sphere of
Uq(M). The manifold M is said to be a Blaschke manifold if it is a Blaschke manifold at

every point in M.

The CROS Ses are examples of Blaschke manifolds and the exotic spheres and exotic
quaternion projective planes are examples of pointed Blaschke manifolds (of. Besse,
[ Be.A ], p. 143).

Blaschke conjectured that any Blaschke manifold is isometric to a CROSS. For
the case that the dimension of the manifold is two, L.W. Green, [ G ], proved that the
conjecture is true : If M is a two-dimensional Blaschke manifold, then M is isometric to a

standard sphere $2 or a real projective plane RPZ. It is still remains open when the

20

dimension of the manifold 2 3.

According to Besse, [Be. A], p. 137, a pointed Balschke manifold is characterized

by the spherical cut-locus:

Theorem 0.1. For a Riemannian manifold M and a point p in M, M is a Blaschke

manifold at p if and only if Cut(p) is spherical.

Throughout this dissertaton, all the manifolds and geometric quantities are C°°

unless it is stated otherwise.

 

CHAPTER 1

SURFACES OF A EUCLIDEAN SPACE WITH HELICAL GEODESICS
THROUGH A POINT

§1. Some fundamental concepts.

K. Sakamoto, [ S-2 ], [ S-6 ], K. Tsukada, [ Ts ], and others studied helical
immersions into a Euclidean space or a unit sphere. Here, we recall the definition of the
helical immersion. Let M be a connected Riemannian manifold and x : M —> M an
isometric immersion of M into a Riemannian manifold M. If the image XO‘Y of each
geodesic y in M has constant Frenet curvatures which are independent of the choice of the
geodesic y, then x is called a helical immersion. K. Sakamoto classified submanifolds of a
Euclidean space or a unit sphere according to the case of even order or odd order helical
immersion.

On the other hand, B.-Y. Chen and P. Verheyen, [ Ch.B-V—l ], [ Ch.B-V-2],
introduced the notion of submanifolds with geodesic normal sections in a Euclidean space
and they completely classified surfaces with geodesic normal sections in a Euclidean space
Em (3 S m S 6). P. Verheyen, [V], showed that submanifolds with geodesic normal
sections in a Euclidean space are equivalent to the helical immersion of the submanifold into
a Euclidean space. We now recall the definition of a normal section. Let M be an n-
dimensional submanifold of a Euclidean space Em of dimension m. Let p be a point of M
and t be a nonzero vector tangent to M at p. Let E(p ; t) be the affine space generated by t

and normal space TgM at p. Then the dimension of E(p ; t) is m-n+1. The intersection of

M and E(p ; t) gives rise to a curve on a neighborhood of p. Such a curve is called the

21

22

dermal eeep'en of M at p in the direction of t. We say that the submanifold M has gepdesi_c
nermel seep'ene if every normal section is a geodesic.

If all the geodesics in a submanifold, regarded as curves in the ambient Euclidean
space, are helices of the same Frenet curvatures, then the submanifold is characterized as an
n-dimensional plane or a Blaschke manifold ([ S-6 D. For surfaces in a Euclidean space
Erlrl (3 S m S 6) with such property, they are planes, standard spheres or Veronese surfaces
which lie in E5 ([ Ch.B-V-2 D. For later use we introduce the Veronese surface in E5 : Let
(x, y, 2) be the standard coordinate system of E3, 3-dimensional Euclidean space, and (ult
u2, u3, u4, u5) be the standard coordinate system of E5. We consider a mapping defined
by

V?” vs (3

u4 =—1--;-(x2 - y2), u5 = %(x2 + y2 - 222).

21/—

This defines an isometric immersion of 8205) into 84(1). Two points (x,y,z) and (-x, -y,
-z) of 820/3) are mapped into the same point of 84(1), and this mapping defines an
imbedding of the real projective plane RP2 into 84(1). This real projective plane irnbedded
in 84(1) is called the Verenege surfeee. It is a minimal surface of S4( 1).

On the other hand, it is well-known that a helical immersion is K-isotropic. We now
recall the definition of isotropy. An isometric immersion x : M —> Em is said to be R.-
isotropic at a point p if 7t = II o(X, X) H does not depend upon the choice of the unit vector
X tangent to M at p. If A is also independent of the choice of point, then x is said to be

conetant isotropic. It is easily seen that M is 7t-isotr0pic at p if and only if

S Agog Y) z = A25 <x, Y>Z

23

for every X, Y, Z 6 Tp(M), where S denotes the cyclic sum with respect to X, Y, Z.
B. O'Neill, [ On ]. Proved that M is isotropic at p if and only if

(1.1) <G(X, X), C(X, Y)> = 0

for any two orthonormal vectors X and Y in Tp(M).

Using O'Neill's idea, we can prove the following.

Lemma 1.1. Let T be a symmetric tensor of type (0, r) defined on E“. Then

|| T(u") ll does not depend on the choice of the unit vector u if and only if
(1.2) <r(ur), T(ur-l, ui)> = 0
for any vector uJ‘ which is perpendicular to u, where T(ur) = T(u, u, u,..., u) and T(ur'l,
ui) = T(u, u,..., u, u‘L ).

Proof. Since all the unit vectors in Em form a unit sphere, a vector ui which is
perpendicular to a unit vector u is tangent to the unit sphere. So,

(1‘ (ur), T(ur)> = C(constant) on the unit sphere

if and only if

uJ'<T(ur), T(ur)> = 0.

Since u is a position vector on the unit sphere,

24

<T(ur), T(ur-l, u‘L)> = 0 (Q. E. D.)

Throughout this dissertation, tJ‘ always means a unit vector perpendicular to t for

some vector t unless it is stated otherwise.

Lemma 1.2. Let M be a submanifold in a Riemannian manifold M such that M

is isotropic at a point p in M. Then we have
(1.3) || 6(e1, e1) II2 = < o(e1, e1), 0(e2, e2) > + 2 ll o(e1, e2) II2

for every pair of orthonormal vectors eland e2 tangent to M at p.
el+ez

\5 .

Proof. Let e1 and e2 be orthonormal vectors tangent to M at p. Set X =
Y = 11-32 . Then X and Y are orthonormal. Since M is isotropic at p,

{2'

<o(er,e1), 0(er,e1) > =< 0(X.X). c(X, X) >.

Using (1.1), we obtain (1.3). (Q. E. D.)

§2. Surfaces in Em with property (*1)

From now on, we assume that M is a complete connected surface in Em(m 2 3)

with Riemannain connection V. We also denote the normal connection by D and the

Weingarten map associated to a normal vector E, by Ag and the second fundamental form by

G as usual.

We now define the property (*1).

25

(*1) : There is a point p in M such that every geodesic through p, which is regarded

as a curve in E“, is a helix of the same constant Frenet curvatures.

Clearly, every helical immersion satisfies the property (*1).
Suppose M has the property (*1). Since every geodesic has the constant

curvatures, < c(t, t), 0(t, t) > does not depend on the choice of the unit vector t e TpM.

So, M is isotropic at p.

Lemma 1.3. Let M be a surface in a Euclidean space Em (m 2 3). Suppose that M

satisfies the property (*1). Then M is isotropic at p.
We now prove

Theorem 1.4. Let M be a complete connected surface in E3. Then M satisfies the
property (*1) if and only if M is a standard sphere or a plane E2.

Proof. Suppose that M satisfies (*1). By Lemma 1.3 we see that M is isotropic at
p. In this case, p is an umbilical point. Choose a geodesic 7 through p. Suppose y is of
rank 1. It is clear that M is a plane E2 in E3. Suppose that 'y is of rank 2. Since every
geodesic is a circle of the same radius and the same center, M is a standard sphere.
Suppose that 'y is of rank 3. We assume that y is parametrized by the arc length s. Let Y(s)
= T. Then 'y"(s) = 6(T, T) and Y"(s) = "A0(T,T)T + (§T6)(T, T) since yis a geodesic.
Since Y is of rank 3, 'y'(s) A T(s) A Y"(s) at 0, and so T A 6(T, T) A AG(T,T)T at 0
along 7. It follows that T A Ao(T,T)T ¢ 0. Since M is isotropic at p, <o(t, t), 6(t, ti)>
= 0, where t = T(O) and y (0) =p. Accordingly, Acmt 1 ti, i.e., Acmt A t= 0. So,

this case cannot occur. The converse is clear. (Q. E. D.)

26

We now assume that a surface M which lies in Em (m 2 4) is compact and
connected.
Suppose that M satisfies the property (*1). By Lemma 1.3 M is isortropic at p.

The equation (1.3) implies that only two cases may occur:

Case 1) 6(e1, e2) 5 0 for any orthonormal vectors el and e2.
In this case, dim(Im (5)1, = 1 since M is compact, where (Im (7)1, ={o(X, Y) | X, Y e TpM}
is called the first normal space at the point 2.

Case 2) 6(e1, e2) at: 0 for an orthonormal basis {e1, eg} of TpM. In this case, dim

(Im 0‘)p 2 2.

Lemma 1.5. Let M be a compact connected surface in E4 satisfying the property
(*1). If the dimension of the first normal space at p is one, then M is a standard sphere
lying in E3.

Proof. Suppose dim (Im 0);, = 1, i.e., 6(e1, e2) = O for any orthonormal basis
{e1, e2} of TpM. So, p is an umbilical point of M. Choose a geodesic 7 through p such
that ’Y (O) = p and 'Y '(s) = T. Then we have

(1.4) Y "(s) = o(T, T),
(1.5) Y "'(s) = -AG(T,T)T + (vToxT, T),

-2A (510)“, T)T +DT((vTo)(T, T)).

27

where (YTA)T1 Y = VX(An Y) - ADXnY - AnVXY for x, Y 6 TM and n 6 TM.

We are going to show that 'Y is of rank 2. It is enough to show that (VtoXt, t) = 0,

where T(O) = t.
Suppose that (YtoXt, t) ¢ 0. We may put

0(T, T) = K15.
where K1 is the first Frenet curvature of 'Y and E a unit normal vector field along 7. As a

matter of fact a is in the direction of the mean curvature vector H at p. Since K1 is

constant,
< (fiToxT, T), o(T, T) > = 0 along Y
and since 6(t, H) = 0, we get
A(YIOXL t)t = 0.

On the other hand, K1 = < 0(T, T), o(T,T) > = < AO(T T)T’ T >. Covariant

differentiation of this equation along the geodesic Y leads to

< (VTA)O.(T,T)T, T > = 0

because Y is a geodesic. Evaluate this at p and then we have

 

28

< (VtA)§t, t > = 0

since 321% 0. Since this holds for any direction, linearization and the Codazzi equation

(0.11) give

(Wag = 0.
So, we can obtain the following :
Y '(O) = t,
Y"(O) = o(t, t),
Y "'(0) = 43.60, wt + (€7,ng t),

y<4>(o) = -o(AG(t, 0t, r) + Dt((t—7To)(r. T)).

Since the curvatures of Y are constant, < Y "(0), Y "'(O) > = O and < Y "'(0),

Y(4)(0) > = 0. Since Yl4)(0) is a normal vector to M, Y"(0) A Y(4)(0) = 0. Thus, 7 is of

rank 3. Since M is compact, this is impossible. Therefore, we have

(Ytoxr, r) = 0.

Since the curvatures are constant, (V_7T0')(T, T) = 0 along 7. So, 7 is of rank 2.

Thus every geodesic through p is of rank 2. Moreover, every geodesic through p is a circle

29

of radius .1— and centered at p — ..l—g and so M is a standard sphere 8201—), (Q, E. D.)
K1 K1 K1

Suppose dim (Im 6);, = 2. Then there is an orthonormal basis {e1, e2} of TpM

such that C(el, eg) at 0.

Lemma 1.6. Let M be a surface in Em (m 2 4) such that M is isotropic at p,

where dim (Im o)p = 2. Then |l o(e1, e2) ll does not depend on the choice of the

orthonormal basis {e1, e2} of TpM.
Proof. Let {X, Y} be an orthonormal basis of TpM. Then there exists 9 (0 S 0 <

21):) such that

X=cosGe1-sin9e2,

Y=sin9e1+cosee2,
for the orthormal basis {e1, e2} of TpM.
Since M is isotropic at p, 0(e1, e1) .1. 0(e1, e2) and 0(e2, e2) .1. 6(e1, e2).
So, o(e1, e1) A 6(e2, e2) = 0 because dim (Im (3)1, = 2. Since || o(e1, e1) II = II 0(e2, e2) II,
C(el, cl) = i 0(e2, e2). If we observe (1.3), then we obtain
(17) 0(61, el) + 6(62, 62) = 0.
that is, the mean curvature vector H vanishes at p. Therefore, we get

O'(X, Y) = COS 29 6(61, 62) + Sin 29 6(61, 61).

If we compute the length of 0(X, Y) and make use of (1.3), then we see that

30

|| 6(X, Y) II = II C(el, e2) ll (Q. E. D.)

In this case, we are also going to prove that every geodesic through p is of rank 2.
Suppose (YtoXt, t) :6 O forte UpM.

Let {e1, e2} be an orthonormal basis for TpM. Consider a geodesic Y1 such that

 

Y1(O) = p, Y1’(0) = 614-62. Since Y1 has its constant first Frenet curvature , we have

«172

<(YTloxrr, T1), 0(T1.T1)> = o.
le . .
where T1 = E. Smce the mean curvature vector 18 zero at p, we see that
(1.8) < (Yeloxer. 61). c(er, an > + 3 < (Yeloxter, e2), 0(61. e2) >

+ 3 < (Velmez. 62). 0(61, 62) > + < (income 62), 6(61, 62) > = 0.

e1-e2

\1—2

 

Consider another geodesic Y2 such that Y2(O) = p, Y2’(O) = . Then we get

< (YTZGXTz. T2), 6(T2. T2) > = 0,
where Y2’(s) = T2. This implies
(1.9) -< (Yeloxer, 61), 0(61, co > + 3 < (Yeloxeheo, 0(61.62) >

- 3 < (Yelmez, e2). 0(61, e2) > + < (Vezoxtez, e2). 0(61, co) > = 0.

31

Putting (1.8) and (1.9) together, we obtain

(1.10) 3 < (Veloxer, e2), 6(61, «22) > + < (Yczoxtez. as). 0(61.62) > = 0.

On the other hand, since geodesics through p have the same constant curvatures,

<(Yo)(X, X, X), (V70)(X, X, X) > is independent of the choice of the unit vector X. By

Lemma 1.2, we have
< (Yoxx, X, X), (Yoxx, x, xi) > = 0

for every unit vector X tangent to M at p. So, (YoXel, e1, e1)J_(V_70)(e1, e1, e2).

Since (€70)(e1, e1, e1) .1. 6(e1, e1), we get
Waxes 61.62)J- 6(61. e2).
Therefore, (1.10) implies that
<(Yczo)(e2, e2), o(e1, e2)> = 0.

Since (VezoXeg, eg) A 0(e1, e2) = 0, we obtain

(V7620X62, e2) = 0.

But, this contradicts (Ytoxt, t) at 0 forte UpM. Thus, it follows that (Ytoxt, t) = 0 for

every t e UpM, i.e., every geodesic through p is of rank 2. By using the fundamental

32

theorem of curves, we can write the immersion x :M —) E4 with respect to the geodesic

polar coordinate system (s, 9) as

(1.11) X(s, e) = C(e) +-]-1<-(cos Ks)f1(0) + ism Ks)f2(6),

where C(O) is a vector function depending upon 6, f1(6) and f2(6) are orthonormal
vectors in E4 at p depending on 9 and K is the Frenet curvature of each geodesic through p.
Without loss of generality we may assume the point p is the origin 0 of E4. Then

we have

(1.12) 0 = x(0, e) = (2(9) +if1(0) for all e.

Let el and e2 be orthonormal vectors tangent to M at 0 which generate the geodesic
polar coordinates (s, 6).
Since x...(a/Bs)(0, 9) = f2(6) e TOM,

(1.13) f2(0) = cost) el + sin 6 e2.
Since ( {7 x (8/83))(0 6) = a_2_x_ (0 0) = 0(f (6) f (9))
Kala/as) * ’ 382 a 2 r 2 a

(1.14) 662(9). me» = - 71-11(6).

where V is the Riemannian connection in E4.

Combining (1.12), (1.13) and (1.14), we obtain

33

x(s, e) = i—(sin Ks)f2(9) + £5 (1 - cos Ks) (562(9), f2(9))

sin KS cosOe1+-l-sinKs sin9e2
K

KIH

+ %(1 - cos Ks)cos 26 6(e1, e1) + L20 - COS KS)Sin 29 6(61! 62)
K K

since 6(e1, el) + o(e2, e2) = O.
0(61.61) = 6(61,61)
||0'(e1,e1)|| K

 

Since 6(e1, e1) .L 0(e1, e2), choose eg, as and e4 as

 

6(61.62) = 0(61.62)
Ilo(e1,e2)ll K

If we use the coordinate system with respect to e1, e2, e3 and e4, then x(s, 9)

becomes

(1.15) x(s, 6) = (i-Sin KS cos 6, l-sin KS sin 9, 1—(1 - cos KS)COS 26,
K K

1— (1 — cos Ks)sin 29).
K

We now prove

Lemma 1.7. Let M be a compact connected surface in E4 . Suppose that M

satisfies (*1) and that dim (Im 0)}, is maximal. Then M is a Blaschke surface at p and M is
diffeomorphic to a real projective space RP2 but not isometric to RP2 with the standard
metric.

Proof. In order to apply Theorem 0.1, it is enough to show that the cut-locus

Cut(p) of the point p is spherical.

34

We may assume that p is the origin 0 of E4. Since each geodesic through 0 is a
circle of radius %, we have to show that two distinct geodesics through 0 do not intersect

on the open interval (0, g). Suppose x(s, 6) = x(so, 60) for 0< s, so < E and 0 <19 — Ool

< 12-3. By using (1.15), we can easily derive a contradiction. Thus Cut(o) is spherical.

Cut(o) is indeed the set of all antipodal points of o with respect to each geodesic through 0.
According to Bott-Samelson, [ Bo], [ Sa ],(or 7.33 Theorem of Besse, [ Be.A D,
we see that M is diffeomorphic to RPZ. We now prove that M is not isometric to RPZ. It is
sufficient to show that the Gaussian curvature K cannot be constant.
Suppose that the Gaussian curvature K is a constant > 0. Then we can easily get G

= % sin2(\ff s), where G = < x*(a/89), x*(a/89) >. On the other hand, G can be directly

computed from (1.15) as

G=—1-i-{ sin21<s+4(1-cos1<s)2 } .
it

If we compare these two equations, we have a contradiction. So, even though the surface is

diffeomorphic to RPZ, it is not a standard real projective space RPZ. (Q. E. D.)

Conversely, if a compact connected surface M is a standard sphere 82 or has the
form of ( 1.15), then it is easily proved that M satisfies the property (*1).
Thus by combining Lemma 1.5, Lemma 1.7 and the statement above we can

conclude the following

Theorem 1.8 (Classification). Let M be a compact connected surafce in E4. Then
M satisfies the property (*1) if and only if M is a standard sphere which lies in E3 or a

Blaschke surface at a point which lies in E4 of the form ( 1.15). In the second case M is

35

diffeomorphic to RPZ.

Remark. In the case of the Blaschke surface at the point 0 in the above theorem,

we see that the locus of the centers of geodesics through the point 0 is a circle with radius K

whose points go around the circumference twice while points on a geodesic circle centered
at 0 on M go around the geodesic circle once. On the other hand, if we compute the torsion

of the cut-locus Cut(o) = x(rt, 9) of the point 0, then we see that the torsion is zero and

Cut(o) is indeed a circle.

Let us consider a compact connected surface M in E5. We shall characterize
surfaces in E5 satisfying the property (*1).
We now suppose that M satisfies the property (*1). As usual, we may assume the

base point p of (*1) as the origin 0 of E5. Then the immersion x : M —) E5 can be

expressed in terms of the geodesic polar coordinates (s, 6) as

(1.16) x(s, 9) = r1(cos Bs-1)f1(9) + rlsin [3s f2(9) + r2(cos 88-1)f3(6)

+ r2 sin 58 f4(9),

where r1 and r2 are nonnegative numbers, [3 and 5 are some positive constants and f1(6),

f2(6), f3(6) and f4(9) are orthonormal vectors in T0E5 depending on 9.

Without loss of generality, we may assume

(1.17) f1(0) = (1, 0. 0. 0. 0). f2(0) = (0, 1, 0, 0, 0), f3(0) = (0. 0, 1, 0, 0).

f4(0) = (0, 0, 0, 1, 0).

36

Let f5(9) be a unit vector in T0E5 such that {f1(9), f2(9), f3(9), f4(9), f5(9)}

forms an orthonormal basis for T0E5. Automatically we may set

(1.18) f5(0) = (O, O, 0, O, 1).

Differentiating (1.16) with respect to s for a fixed 9, we obtain

(1-19) Kala/38) = 4113 sin BS f1(9) + r113 COS [38 f2(9) - r25 sin 58 f3(9)
+ r25 cos 8s f4(6).
Set
(120) 6(9) = X*(3/38)(0. 9) = r113 f2(9) + r25 f4(9).
which implies
(121) (I192 + (r25)2 = 1-
Let

e1= e(O) = r1B f2(0) + r25 f4(0) = (0, r1B, O, r25, 0)

and

37

1t 1t 1t
62 = 6(7) = nil f202") + r25 My).

Then e(9) can be expressed as

(1.22) e(6) = cos 9 el + sin 6 e2.

For a fixed 9, x(s, 9) IS a geodesic and thus< 3—3 ,3—2 >= 0, which gives

(1.23) rfa < me), f2(6) > + r3 5 < 13(9), f4(6) >

+ ( -rfB < me), 15(9) > + r1r2B < f2'(9). W» > ) cos B8
- ( .r35 < 13(9), we) > + r1r25 < f1'(9)a we) > ) cos 8s
- rlrgfl < me), f3(6) > sin [35 + r1r25 < me), f3(6) > sin 5s
- 323(1) - 5x < rite). 14(9) > + < me). f3(9) > ) cos (B + 8)s
+ r1_2r2_ (a + 5)( < fire), we) >- < f2'(e). f3(9) > > cos (B -5>s
+r—-12r2 —(B- 5)( < f1'(0), f3(6) > < 12(9), f4(e) > ) sin(B + 5)s

+1294) +5)( < 11(9), f3(0) > + < me), t4(e) > ) sin(B - 8)s

38

Lemma 1.9. B at 8 provided the geodesics through 0 are of rank 4.
Proof. Suppose B = 8. Let K1, K2 and K3 be the first, second and third Frenet

curvatures of the geodesic x(s, 0) for a fixed 0 respectively. Then

2 2
K? = < 33'“), 9), %(0, 9) > = (If +r§)B4.

Since <g)s-‘(0, 0), $30, 0) > = 1, we obtain
(r'f+r§)132 = 1,

1

 

that is, B2 = 2 2 . Therefore, K1 = B.
r1 '1' r2

On the other hand, the curvatures Ki's and the frequencies B and 8 have the

following relations :

Kf+K§+K§=BZ+67~=2B2,

Kng: [5252:54.

Since K1 = B, K; = B2. The first equation gives K2 = 0. This contradicts the fact

that x(s, 0) is of rank 4. Thus, we have B :6 5 . (Q. E. D.)

Lemma 1.10. For every 0, the geodesic x(s, 0) is periodic.

Proof. If x(s, 0) is of rank 2, then this is obvious. We now assume that x(s, 0)

39

is of rank 4. Suppose x(s, 9) is not periodic. Then B and 8 are independent over the
rational numbers, that is, x(s, 9) = Sl(r1) x Sl(r2), a torus denoted by T, where x(s, 9)
is the closure of x(s, 9) in E5. Certainly, T is contained in x(M). But T does not satisfies

the property (*1). Thus x(s, 9) must be periodic for every 9. (Q. E. D.)

We now suppose that r1 ¢ 0 and r2 :6 0, that is, every geodesic through 0 is of rank

4. Combining (1.23), Lemma 1.9 and Lemma 1.10, we obtain
(1.24) < f1'(9), f2(9) > = < f1'(9). f3(9) > = < f1'(9). f4(9) >
= < f2'(9). f3(9) > = < f2'(9). f4(9) > = < f3'(9), f4(9) > = 0

for all 9. So we have the following system of differential equations :

4
(1.25) me) = Home) . me) = Exits) me)
i=1

for i = 1, 2, 3 and 4, in other words,

(mam ( 0 0 o 0 21(e)\(f1(9)\
me) 0 0 0 0 22(9) 12(9)
f3'(6) = 0 0 0 0 23(9) f3(9) .

f4'(9) 0 O O 0 14(9) f4(9)
(5(9)) Hum-12(9)mum-14(9) 0 Ana»)

 

 

 

 

 

 

where the Ms are periodic functions with period 27:.
Differentiating (1.20) with respect to 9 and making use of ( 1.22) and (1.25), we

get

40

(1.26) (r1B A2(9) + r25 M(9))f5(9) = -sin 9 61 + COS 9 62,

from which, we obtain that r1B M(O) + r28 24(9) = i1 and we may assume that r1B 12(9)

+ r25 14(9) = 1. If we differentiate (1.26) twice and use (1.25), then we obtain
f5"(9) + f5(9) = 0

Since f5(0) = (0, 0, 0, o, 1) and 13%) = -e1= (o, -r1B, 0, -r28, 0), we have

(1.27) f5(9) = (0, -r1B sin 9, 0, 425 sin 9, cos 9).

Since fi'(9) = ki(6)f5(6) (1 s i s 4). mm = (1, 0. 0, 0. 0). mm = (0, 1. 0. 0. 0). f3(0)
= (0,0, 1, 0,0) and mm = (0, 0, 0, 1, 0), we get

9 9 9

(1.28) me» = (1, 41136910) sin t dt, 0, 42801210) sin t dt, IMO) cos tdt),
0
9 9 9
(1.29) 12(9) = (0, -r1B6[7»2(t) sin t dt + 1, 0, 4250920) sin t dt, Upon) cos t dt),
9 9 9
(1.30) f3(0) = (0, -r1[iojx3(t) sin t dt, 1, -r25JA3(t) sin t dt, Jim) cos t dt),
9 9 9

(1.31) 14(9) = (0, -r1[36[>t4(t) sin t dt, 0, 42869.40) sin t dt + 1, Jinn) cos t dt).

41

Now let us compute 21(9). Since < f1(9), f5(9) > = 0 for all 9, (1.27) and (1.28)

imply

9 9 9
(rlB)2 sin 9 Illa) sin tdt + (r28)2 sin 9 69.10) sin tdt + C089 01110) cos t dt = 0.
0

It follows that

9 9
sin 9 69.16) sin t dt + 0039 OIMQ) cos t dt = 0

because (r1 B)2 + (r2 5)2 = 1. By differentiating this, we obtain

9 9
31(9) = -cos 9 Jim) sint dt + sin 90pt1(t) cos t dt,

which gives M'(9) = 0 for all 9, that is M is a constant, In fact, 11(9) = 0 for all 9.

Thus, f 1(9) is completely determined :
f1(9)=(1, 0, 0, 0, 0).
Similarly, we can compute M, A3 and A4 :
12(9) = r1B, A3(9) = 0, 14(9) = r28 for all 9.

Consequently (1.28) ~(1.31) are precisely determined as follows :

42

f1(9) = (1, O, O, 0, O),
f2(9) = (0, 1 - (rlB)2(1 - cos 9), 0, -r1r2 B8(1 - cos 9), r1B sin9),
f3(9) = (O, 0, 1, 0, O),
f4(9) = (1, -r1r2 B8(1 - cos 9), 0, 1 - (r28)2(1 - cos 9), r28 sin9).
These, together with (1.27), show that the immersion x the representation
(1.32) x(s, 9) = (r1 (cos Bs - 1), r1 sin Bs - r1B (1 - cos 9)(r%B sin Bs + r3 8 sin 85),
r2 (cos 8s - 1), r2 sin 83 - r28 (1 - cos 9)(r%B sin Bs + r38 sin 85),

(r%B sin Bs + r38 sin 85) sin 9).

2 2
In this case, each geodesic through 0 is periodic with period L = 12- : £3- for

some integers p and q. Using a similar argument to that in Lemma 1.7, we see that the
cut-locus, Cut(o), of the point 0 is spherical and thus the surface M is a Blaschke manifold

at 0 which is differemorphic to RP2. Thus, we have

Proposition 1.11. Let M be a compact connected surface in E5 with the
property (*1) relative to the origin 0. If every geodesic through the point 0 is of rank 4,
then M is a Blaschke manifold at 0 which is diffeomorphic to RP2 and has the form (1.32).

We now suppose that x(s, 9) is of rank 2 for every 9. Then the immersion x can

43

be written with respect to the geodesic polar coordinate (s, 9) as

(1.33) x(s, e) = i- (cos KS - 1) 11(9) +-]1;sin KS 13(9),

where K is the Frenet curvature of the planar geodesic x(s, 9) for every 9 and f1(9) and

f2(9) are orthonormal vectors in E5 at the point 0. From (1.33) we obtain

(1.34) x*(a/as)(0, 9) = f2(9) e TOM

and

... 2
(1.35) (Vx*(a,a,)x.<a/as>)<0. e) = 38—; (0.9) = caste), f2(9)) = -t< tr (e),

where V is the Riemannian connection in E5.

Let {e1, e2} be an orthonormal basis of TOM such that
(1.36) f2(9) = cos 9 el + sin 9 e2.

Suppose that dim (Im (5)0 = 1, that is, the point 0 is an umbilical point of M . In

this case, every geodesic through 0 is a circle of radius land centered at - l- H. Thus, M is
K K

a standard sphere SZ(-1-) which lies in E3.
K

Suppose that dim (Im (5)0 = 2. In this case, exactly the same proof used to derive

(1. 15) is applied and thus the immersion x is of the form

(1.37) x(s, 9) = 1- (sin Ks cos 9, sin Ks sin 9, (1 - cos Ks) cos 29, (1 — cos Ks) sin 29, 0)
K

for a suitable choice of Euclidean coordinates in E5. Clearly, M lies in E4.

We now assume that dim (Im (5)0 = 3, that is, the dimension of the first normal

space at the point 0 is maximal. Then

0(61. 61) A 0(61, 62) A 0(62. 62) ¢ 0.

(1.38) e3 =M , arc—((115132)- and (is-1%,
K "65”

g(0(61. 61), 0(62, 62))

2 6(e1, e1). Set b =
K

where a = II 0(e1, e2) II and 85 = 0(e2, e2) -

ll 65 ll. Then we have from Lemma 1.2 that

(1.39) b2 +

Using (1.34), (1.35), (1.38) and (1.39), we can write the immersion x in the form

1 . 1 . . 1 2a2 . 2
(1.40) x(s, 9) = (— srn Ks cos 9, -- srn Ks sm 9, -—(1 - cos Ks) (K - — sm 9),
K K K2 K

2.

(1 - cos Ks) sin 29, -b-(1 - cos Ks) sin2 9)
K2 K2

for a suitable choice of the coordinates with respect to e1, e2, e3, e4 and e5 described

above.

45

Considering the cut-locus Cut(o) of the point 0 in both the cases that dim(Im (7)0 =

2 and dim(Im (5)0 = 3, we see that M is a Blaschke surface at 0.

Conversely, if the immersion x has the form (1.32), (1.37) and (1.40) or x is a
standard imbedding of 82(1) into E3, then it is easily checked that the surface M satisfies
K

the property (*1).

Thus we can classify surfaces in E5 satisfying the property (*1).

Theorem 1.12.(Classification). Let M be a compact connected surface in E5.
Then M satisfies the property (*1) if and only if M is a standard sphere in E3 or a Blaschke
surface at a point of the form (1.37) which lies in E4 or a Blaschke surface at a point of the
form (1.40) which lies in E5 or a Blaschke surface at a point of the form (1.32) which lies
in E5 . All such pointed Blaschke surfaces are diffeomorphic to RPZ.

Remark. Let M be a compact connected surface in Em (m 2 5). Since the
dimension of the first normal space at a point is at most 3, we can conclude that M satisfies
the property (*1) and the geodesics are planar if and only if M lies in E5 and M is one of
four model spaces stated in Theorem 1.12 except the case of a Blaschke surface of the form

(1.32).
§3. A new characterization of the Veronese surface

The Veronese surface introduced at the beginning of this chapter certainly satisfies
the property (*1). So the following question naturally arises : What is the characterization
of the Veronese surface in terms of the property (*1) ? Since the Veronese surface is fully
immersed in E5, that is , the Veronese surface cannot lie in a hyperplane of E5, and since

every geodesic in the Veronese surface is planar, we must think of the immersion which

46

has the form (1.40).

We are going to use the theory of submanifolds of finite type introduced and mainly
developed by B.-Y. Chen, [ Ch.B-3 ]. We recall some fundamental definitions and
properties.

Let M be a compact orientable Riemannian manifold with Riemannian connection V

and A the Laplacian operator of M acting on C°°(M), where

A = -2(VEiVEi ' VVEiEi)
i

for an orthonormal basis { Ei } of TM. We define an inner product ( , ) on C°°(M) by

(f, g) = jfg dV,
M

where dV is the volume element of M. Then A is a self-adjoint elliptic operator with

respect to ( , ) and it has an infinite, discrete sequence of eigenvalues :
0=7to< 71.1< 12< <7tk< T +oo.

Let Vk = {f e C°°(M) | Af = kkf } be the eigenspace of A with eigenvalue M. Then ZVk
k=0

is dense in C°°(M) in the L2 - sense. Denote by ék Vk the completion of XVk. We have
k=0

C°°(M) = ék Vk.

For each function f e C°°(M), let ft be the projection of f onto the subspace Vt (t =

47

0, 1, 2,... ). Then we have the spectral decomposition

f = 2 ft (in the L2 -sense).
t=0

Because V0 is l-dimensional, for any non-constant function f e C°°(M), there is a
positive integer p 2 1 such that fp :6 0 and

f‘f0= 2 ft,
t2P

where f0 6 V0 is a constant. If there are infinitely many ft's which are nonzero, we put q =

+00. Otherwise, there is an integer q, q 2 p, such that fq at 0 and

If we allow q to be +oo, we have the decomposition as above for any f e C°°(M).
For an isometric immersion x : M —> Em of a compact Riemannian manifold M into

Em, we put

X = (X1, X2,..., Xm),

where X), is the A-th Euclidean coordinate function of M in Em. For each xA, we have the

spectral decomposition

CIA
XA- (xA)0= 2 (XA)t. A = 1. 2,.... m.
t=PA

For each isometric immersion x : M —> E“, we put

48

P=P(X)=ian{ PA }, q=q(x)=supA {qA }.

where A ranges among all A such that xA — (xA)o 96 0. It is easy to see that p is an integer

2 1 and q is either an integer 2 p or co. Moreover, p and q are independent of the choice of
the Euclidean coordinate system in E“. Thus p and q are well-defined. Consequently, for
each isometric immersion x : M —> Em of a compact Riemannian manifold, we have a pair

[p, q] associated with M. We call the pair [p, q] the erder ef the submanifeld M. If we

use the spectral decomposition of the coordinate functions of the immersion x : M —> E“,

we have
q
(1.41) x=x0+ 2‘, xt.
t=p
Definition. A compact submanifold M in Em is said to be of finite type if q is

finite. Otherwise M is of infinite me.

Definition. A compact submanifold M is said to be of k-type (k = 1, 2, 3,...) if

there are exactly k nonzero xt's (t 2 1) in the decomposition ( 1.41).

We can restate Ilfakahashi's Themem in terms of l-type :

Lemma 1.13 (Takahashi [ Tk ] and Chen [ Ch.B-3 ]). Let x : M —9 Em be an
isometric immersion of a compact Riemannian manifold M into 13‘“. Then x is of 1-type if

and only if M is a minimal submanifold of a hypersphere of E“.

B.-Y. Chen gave the following characterization of submanifolds of finite type.

49

Lemma 1.14 (Chen [ Ch.B-2 ]). Let x : M ——) Em be an isometric immersion of a
compact Riemannian manifold M into 13‘“. Then

(i) M is of finite-type if and only if there is a non-trivial polynomial Q such that
Q(A) H = 0.

(ii) If M is of finite type, then there is a unique monic polynomial P of least degree such

that
P(A)H = 0.
(iii) If M is of finite type, then M is of k-type if and only if deg P = k.

Now, coming back to the problem. We shall compute the Gaussian curvature K
and find a condition which gives constant Gaussian curvature. Furthermore, we shall
characterize the Veronese surface by examining the surface with constant Gaussian
curvature and is of 1 -type.

From ( 1.40) we get

x*(a/Bs)(s, 9) = (cos Ks cos 9, cos Ks 3m 9, — srn Ks (K - — sm 9),
K K

a . . b . .
— srn Ks sm 29, —sm Ks s1n2 9).
K K

and

50

- 2
x,,(3/39)(s, 9) = ( - 1— sin Ks sin 9, -1- sin Ks cos 9, ~2—2-(1 - cos Ks) sin 29,
K K K

21(1 - cos Ks) cos 29, -b— (1 - cos Ks)sin 29).
K2 K2

Then the induced first fundamental form gij is derived as
311 = < x*(3/3S). X*(3/38) > = 1, g12 = g21 = < Kala/BS). M(a/BG) > = 0,

2
g22 = < x,(a/ae), x,,(9/39) > = -%sin 2 Ks + 4:114-(1 - cos Ks)2.
K

Thus the line element dp2 of M in E5 has the form

2
dp2 = ds2 + { l—sin 2 Ks + 4_a_(1 - cos Ks)2} d92.
K2 K4

So the Gaussian curvature K is given by

82/5

1
1.41 K=-——,
( ) «JG 852

2
where G = g22 = —1—sin 2 Ks + 4a_(1 - cos Ks)2.
K2 K4

Suppose that the Gaussian curvature K is a constant. Then (1.41) is equivalent to

2
a VG 49—9? +4 KG2=0.
BS2 as

 

26

By a straightforward and long computation, we have

51

 

 

 

 

1:32 2:, ix #3:) %(-“i2-1><7-i—)}cosxs
+{4%-1)(fi2-+6—:;)(:—2-1)-3:24+12:8a4K}cosZKs
+ $43-23 -1)(3— %)cos3Ks + 2{CE-(3:511;-.1)2(]I:—2-1)cos4t<s=0.

Since 1, cos Ks, cos 2Ks, cos 3Ks and cos 4Ks are linearly independent, we get

 

 

 

 

= o,
(b) 1:32 ‘21:???“ 1 -%>+4—:,3<:i22 - 1><7--:i2(-) =0,
(e) «if-§- - 1) (;}:3+%2-)(f—2- 1) - 3:24 +12? K = o,
(d) gtti: -1>(3 - 5:128) = o.
1 4a2

K
(6) REF-1)2(E-l)=0.

52

From the last two equations, we get

2 2
41.-1:0 or K=K2 or K=§§—.
K2 4
2
If 4i- - 1 =0, then (b) implies
K2
2
(1.42) K457.

2
If K = K2, then (d) implies 2% = 1. It follows that K = K2 and hence K = 0 by ( 1.42).
K

. . . . . 3K2
This rs a contradiction. By a srmrlar argument, K it —4—.

Thus we have

Proposition 1.15. Let M be a compact connected surface in E5 satisfying (*1)
whose immersion is given by (1.40). Then the Gaussian curvature K is constant if and

2
. . . K
only 1f K = 4a2. In thrs case, the Gaussran curvature K = T = a2.

In such a case, the induced metric (gij) looks like

1 0
(1°43) (gij) =(O isin 2 Ks + LU - cos Ks)2)
K2 K2
Using this induced metric, we can compute the Christoffel symbols I‘j}; :
1‘1—0 1‘2—0 rl—lalo—l'
11— , 11— , 22—‘25'S'(0g )—‘KSIHKS,

53

2 l 2 l a 1 .
F22 = 0, F12 = 0, F12 =5 'a_s'(10g G) = 'ESln KS,

where G = -1- sin 2 Ks + $0 - cos Ks)2.
K2 K2

Lemma 1.16. Let M be a compact connected surface in E5 satisfying the
property (*1) whose immersion has the form (1.40). If the Gaussian curvature is constant,

then the Laplacian operator A is given by

(1.44) A=-(—+
S

where G = -1-sin 2 Ks + i—(l - cos Ks)2.
K2 K2

It is well-known that

Ax = ~2H,

where H is the mean curvature vector field of M. Using this equation and computing H

and AH by means of (1.44), we obtain the following

AH-%K2H=0.

According to Lemma 1.14, M is of 1-type and hence M is a minimal submanifold of a
hypersphere of E5.
Let us recall the Calabi's Theorem.

54

Theorem 1.17 (E. Calabi [ C D. Let 2 be a 2-sphere with a Riemannian metric
of constant Gaussian curvature K, and let x : 2 —-> Sn'1(r) c EI1 (11 = 2m +1 2 3) be an

isometric minimal immersion of 2 such that the image is not contained in any hyperplane

of E“. Then

i) The value of K is uniquely determined as,

_ _2_
— m(m+1)r2’

ii) The immersion x is uniquely determined up to a rigid motion of Sn'1(r) and the 11
components of the vector x are a suitably normalized basis for the spherical harmonics of

order m on E.

On the other hand, we can easily check that the Gaussian curvature K cannot be

2
constant if the immersion has the forms (1.32) or ( 1.37) by computing K = - _1__8 ‘16 .

VG BS2

 

So if we apply Calabi's Theorem, we conclude

Theorem 1.18 (Characterization of a Veronese surface). Let M be a compact
connected surface in E5. Then M is a Veronese surface if and only if M has a constant
Gaussian curvatutre and there is a point p which is not umbilical such that every geodesic

through p is a helix of the same curvatures.

In this case, x is the first standard imbedding RP2 into E5 and the second standard

immersion of 2-sphere into S4.

CHAPTER 2

CHARACTERIZATION OF SURFACES OF REVOLUTION IN
A 3-DIMENSIONAL EUCLIDEAN SPACE

Let M be a surface in E3. We now define (*2).

(*2) : There is a point p in M such that every geodesic through p is a normal section

ofM at p.

Suppose M has the property (*2). Let 'y be a geodesic parametrized by the arc

length s and let 7(0) = p. Then we have
Y6) = T.
Y'(S) = 6(T, T),
Y"(s) = -AG(T, T)T + (VTo)(T, T).
Since 'yis a normal section at p in the direction t = T(0), Ao(t, 0t A t = 0, that is,
< o(t, t), o(t, ti) > = 0.

Since this is true for any orthonormal vectors t and ti tangent to M at p, M is isotropic at p

55

56

and p is indeed an umbilical point since the ambient manifold is E3. Since yis a plane

curve, 7(3) A Y'(s) A Y"(s) = 0 for all s e Dom 7. So we can obtain

T A ACCT, T)T A 0(T, T) = O,

which implies
(2.1) < 0(T, T), o(T, Ti) > = 0

along 7.

Without loss of generality, we may assume p as the origin 0 of E3. Since every
geodesic through 0 is planar, we can express the immersion x : M ——> E3 locally on a

neighborhood U of o in terms of geodesic polar coordintes (s, 9) as

(2.2) x(s, 9) = (h(s, 9) cos 9, h(s, 9) sin 9, k(s, 9))

for a suitable choice of Euclidean coordinates of E3, where h and k are differentiable
functions satisfying h(0, 9) = k(0, 9) = 0.
Differentiating (2.2) with respect to s and 9, we obtain two orthogonal vector fields

tangent to M on U

3 ah ah . 8k
(2.3) “(38.) = ( 5: C08 9, 9—8- srn 9, 9—8- ),
(2.4) x*(-§—)=(§P-cos9-hsin9,arisin9+hcos9,Q-l-{-),
99 99 B9 B9

57

where x*(§§)(0, 9) = ( cos 9, sin 9, 0 ).

For a fixed 9, x(s, 9) is a geodesic and we thus have

a 3 8h 2 3k 2
0915), Xs(3§)>=(5:) +(55-) =1.
We may put
(2.5) :_h = cos f(s, 9), ES—k- = sin f(s, 9),
s s

for a smooth function f(s, 9) defined on U satisfying cos f(0, 9) = 1 and sin f(0, 9) = 0
for all 9.

Lemma 2.1. 5%(K(S, 9))2 = 0 on the neighborhood U, where K(s, 9) is the

Frenet curvature of the geodesic x(s, 9) for a fixed 9.

Proof. Let y be a geodesic such that 7(5) = x(s, 9) for some 9. Then we have
(x(s, 9))2 = < OCT. T). o(T. T) >.

where T(s) = x*(aa§)(s, 9). We now compute gs- (K(s, 9))2 :

is:

- <oT,T,oT,T>
239 < > < >

= < Da/ae 0(9/85, a/as), 0(9/95, 8/85) >

58

= < (Va/aeoxa/as, 8/85), 6(8/85, 8/85) >

+ 2 <o(Va/898/85,8/85), 0(8/8S, 8/85) >

= < 0(Va/aS8/89, 8/85), 6(8/85, 8/85) > ( because of the Codazzi equation

and (2.1))

= < D8 /85 6(8/89, 8/85), 0(8/85, 8/85) >
- < o(Va/388/89, 8/85), 0(8/85, 8/85) >
= g—S-<o(8/89, 8/85), 0(8/85, 8/85) >
- < c(a/ae, 8/85), (Va/asoxa/as, 8/85) >
- < o(Va/398/85, 8/85), 6(8/8S, 8/85) >
= - < c(a/ae, 8/85), (Va/asoxa/as, 8/85) > ( because of (2.1) ).
Let 6(8/89, 8/85) = f1(s) N and let 0(8/85, 8/85) = g1(s)N, where N is the unit

vector field normal to M along y and f 1 and g1 are some smooth functions defined along 7.

Then we have

- 8N
(Va/aSoXa/as, 8/85) = g1'(s) N + g1(s) E'

59

(2.1) leads to f1(s) g1(s) = 0 for all 5.
If g1(0) :6 0, then there exists an interval I contained in Dom 7 such that g1(s) at 0

for s e I. So f1(s) = 0 on 1. Thus, we have

< c(a/ae, 8/85), (Va/aSoxa/as, 8/85) > = f1(s) g1'(s) = 0 on 1.

Suppose that g1(0) = 0. Let so = inf {s l g1(s) :6 0 }. If so = 0, then g1(s) :6 0 for
5 > 0 and thus f1(5) = 0 for s > 0. So, f1(s) g1'(s) = 0 for 5 > 0. By continuity,
f1(s) g1'(s) = 0 for s 2 0. If so > 0, then g1(s) = 0 for s < 50. Thus f1(5) g1'(s) = 0 for
s < 50.

If there is some 5 e Dom 7 such that g1(5) = 0, then we keep doing this argument

and thus we get aa—6(K(s, 9))2 = 0 for s e Dom 7. Since this is true for every 9, we have
8
5-9—< 6(8/85, 8/85), 0(8/85, 8/85) > = 0

on U. In other words, the curvature K(5, 9) is independent of the choice of 9 .

(Q. E. D.)

Lemma 2.2. The functions h and k are independent of the choice of 9.
Proof. Differentiating (2.3) with respect to s, we get
82x 82h 82h 82k

(2.6) 5:2- =(§ COS 9, a? sin 9, 'a-S-z- ).

Thus the curvature K(s, 9) satisfies

60

f
(2.7) (x(s, 9))2 = (2— )2.
85

On the other hand, (2.1) gives

82x 8 8x
2.8 < — , — — > = 0,
( ) 852 89 (85 )
which implies
(2.9) _8i8_f = 0.
89 85

By Lemma 2.1, the curvatures do not depend on 9 and so we choose a geodesic
x(s, 9) for some 9 and examine its curvature.
Suppose that K(O, 9) = 0. Let 51 = inf { s I K(5, 9) at 0 }. If 51 = 0, then K(5, 9)

¢ 0 for s > 0. (2.7) and (2.9) imply 5—3;: 0 for s > 0. By continuity, 3—2 = 0 for s 2

0. If 51 > 0 (possibly +oo), then K(s, 9) = 0 for 0 S s <51. Then the inside of the geodesic
circle S1 centered at o with radius 51 lies in E2. In this case, h and k are clearly
independent of the choice of 9.

Suppose that K(0, 9) :6 0. Then we can choose a sufficiently small neighborhood

of o where K(5, 9) at 0. Evidently :_f it 0 and thus 23—; = 0 on this neighborhood.
5

Developing this argument continuously if K(5, 9) = 0 for some 5 at 0, we see that h and k
are independent of the choice of 9 in the neighborhood U because h(0, 9) = k (0, 9) = 0
for all 9. (Q. E. D.)

Since the functions h and k only depend on the arc length 5, (2.2) defines a surface

of revolution around the point 0.

61

Conversely, a meridian of a surface of revolution is always a geodesic and all the
normal sections at the point 0 are geodesics through 0 if M is locally a surface of revolution

with axis of symmetry passing through 0. Thus we have

Theorem 2.3 (Local characterization). Let M be a surface in E3. Then M is
locally a surface of revolution with vertex p (around a neighborhood of p) if and only if

every geodesic through p is a normal section of M at p.

Theorem 2.4 (Global characterization). Let M be a complete connected surface in
E3. Then M is a surface of revolution if and only if there is a point p in M such that every

geodesic through p is a normal section of M at p.

Let M be a Riemannian manifold immersed in a Riemannian manifold 1V1. A curve

0t in M which is regarded as a curve in M is called a W-curve if the Frenet curvatures of 0t
are constant along 0t. In this case, the Frenet curvatures may depend on the choice of
geodesics. Together this notion and the property (*2) give another characterization of

submanifolds with geodesic normal sections in E3.

Corollary 2.5. Let M be a complete connected surface in E3. Then the
following are equivalent :

(1) M satisfies the property (*2) and the geodesics through the base point in the
property (*2) are W-curves .

(2) M satisfies the property (*1).

(3) M has geodesic normal sections.

(4) M is a plane E2 or a standard sphere S2.

CHAPTER 3

SURFACES OF A EUCLIDEAN SPACE WITH PLANAR GEODESICS
THROUGH A POINT

§ 1. Surfaces of a Euclidean space with planar geodesics through a
point which is not an isolated flat point

Let M be a surface in Em (m 2 3). We define the property (*3).

(*3) : There is a point p in M such that every geodesic through p is planar.

Lemma 3.1. Let M be a surface in Elm and let 7 be a geodesic in M through p.
If 7 is a planar curve, then 7 is a normal section of M at p.

Proof. Let us assume that 7 is parametrized by the arc length 5 and let 7(0) = p.

Then we have
7(8) = T.
Y'(S) = 0(T, T),

7'"(s) = -AO(T, T)T + (VTo)(T, T).

Since 7 is a plane curve, 7(5) A 7"(5) A 7"‘(s) = 0 along 7. Thus we get

62

63

T A o(T, r) A (43.0“, T)T + (VTO'XT, T) ) = 0,

which implies

(3.1) T A ACCT, “T = 0
and

(3.2) o(T, T) A (VTo)(T, T) = 0.

Suppose first that 0(t, t) 96 0, where t = T(0) . We can choose a neighborhood U
of p such that 6(u, u) at 0 for any nonzero vector u e TqM, q e U. So 7 lies in
p + Span{t, o(t, t), (Vto)(t, t)} and hence 7 is a normal section at p.

Suppose 6(t, t) = 0. It is enough to consider O'(t, t) = 0 and 6(T, T) 96 0 for
s > 0. Let N be a normal vector field to M which is parallel to 0(T, T) along 7for s > 0.
Then we can choose a vector field TJ' which is tangent to M along 7 and perpendicular to
the plane II spanned by {T(s), N(s) } (s > 0). Extend TJ‘(s) up to the point p, which will
be denoted by the same notation Ti. Then { t, Ti(0) } is an orthonormal basis for TpM
and Ti(0) is perpendicular to the plane H. N(O) is thus a normal vector to M at p.
Consequently, 7 lies in p + Span {t, N(0) ] and hence 7 is a normal section at p in the

direction t. (Q. E. D.)

Making use of this lemma, we see that the property (*2) is equivalent to the
propertry (*3) if the ambient manifold is a 3-dimensional Euclidean space E3.

Thus we have

64

Theorem 3.2. Let M be a surface in E3. Then, M satisfies the property (*3) if

and only if M is locally a surface of revolution.

Corollary 3.3. Let M be a complete connected surface in E3. Then M

satisfies the property (*3) if and only if M is a surface of revolution.

We now suppose that a surface M in Em satisfies the property (*3). By virtue of
(3.1), we get

(3.3) < o(T, T), o(T, Ti) > = 0,

where 7(5) = T, 7 being a geodesic through p. In particular, < o(t, t), o(t, t‘L) > = 0, t =
T(0). It is true for any unit vetor t in TpM and thus M is isotropic at p. So we may use
Lemma 1.2 later.

Let (s, 9) be the geodesic polar coordinate sytem about p. We may assume that
p is the origin 0 of E“. Let expo(s e(9)) = x(s, 9), where e(9) = cos 9 e1 + sin 9 e2 for
some orthonormal basis {e1, e2} for TOM which is associated with the geodesic polar

coordinates (s, 9).

Lemma 3.4. Let M be a surface in Em with the property (*3). Then

i

89 (K(S, 9))2 = 0.

(3.4)

where K(5, 9) is the Frenet curvature of x(s, 9). In other words, the curvature of each
geodesic through 0 does not depend on 9.

This lemma can be proved similarly to Lemma 2.1.

65

Since every geodesic through 0 is a plane curve and it is a normal section at 0,

we may represent the immersion x : M —) Em locally as

(3.5) x(s, 9) = h(s, 9) cos 9 61 + h(s, 9) sin 9 e2 + k(s, 9) N(9),

where N(9) is a unit vector normal to M at 0 which may depend on 9 and h and k are some

smooth functions satisfying h(0, 9) = k(0, 9) = 0 for all 9.

Lemma 3.5. The functions h and k described as above do not depend on 9.
Proof. Since (5, 9) is the geodesic polar coordinate system, we have the

following orthogonal vector fields tangent to M about 0 :

(3.6) x452) = g:- cos 9e1+:—:- sin9 e2+ 3% N(9),
(3.7) x*(a%.-)=(:—:-cos9-hsin9)e1+(g—2sin9+hc059)e2

8k
+—N0 +kN'9.
89 () ()
. 8 8
Slnce < x*(85)’ x43?) > = 1, we get
8h 2 8k 2
— + — =1,
(as) (as)

from which , we may put

66

(3.8) §1- = cos f(s, 9), 21(- = sin f(s, 9),
8s 85

where f is a smooth function defined on a neighborhood of 0. Since x*(%)(0, 9) =

cos 9e1 + sin 9e2, cos f(0, 9) = 1 and sin f(0, 9) = 0. Using (3.7) and (3.8), the curvature

K is represented as

(3.9) (x(s, (9))2 = ( 23)?
85

On the other hand, (3.3) implies

82x 8 8x
< — , — — > = 0,
852 89 (8s )
which yields
(3.10) if; 8_f = 0
8s 89

The rest of the proof is exactly same as that of Lemma 2.2. (Q. E. D.)

We assume that a surface M lies in E4 satisfying the property (*3) where the base
point p in the property (*3) is not an isolated flat point. An isolated flat point p means a
point such that the curvature of every geodesic through p vanishes only at p in some
neighborhood of p. The curvature tensor R obviously vanishes at flat points. We also

assume that the point p as the origin 0 of E4. Let {e1, e2 } be an orthonormal basis of TOM.

67

Suppose first that dim (Im (5)0 S 1. In this case, by considering Lemma 1.2, we
see that o is an umbilical point of M. If dim (Im (3)0 = 1, then by choosing an appropriate

Euclidean coordinate system of E4 the immersion x : M -—) E4 can be locally expressed in

terms of the geodesic polar coordinate system as

x(s, 9) = (h(s) cos 9, h(s) sin 9, k(s), 0)

for some smooth functions h and k of the arc length 5 due toLemma 3.4, where (s, 9) is the
system of geodesic polar coordinates related to {e1, e2}. Thus M is locally a surface of
revolution about 0 with axis of symmetry in the direction of the mean curvature vector at 0.
Suppose that dim (Im (5)0 = 0. Since 0 is not an isolated flat point, there exists a
neighborhood of 0 which is contained in a plane E2 and this is a special case of above

surface of revolution.

We now suppose dim (Im (5)0 = 2. As we showed in Lemma 1.6, the mean
curvature vector H vanishes at o and II o(t, ti) ll does not depend on the choice of

orthonormal vectors t and ti tangent to M at 0. Choose two unit vectors N1 and N2 normal

to M at 0 such that
(3.11) N1: 6(61,61) and N2: 6(61. 62)
Km) K(O)

where K(0) is the Frenet curvature at 0. Since the functions h and k in (3.5) are

independent of the choice of 9, (3.5) can be reduced to
(3.12) x(s, 9) = h(s) cos 9 e1 + h(s) sin 9 e2 + k(s) N(9).

As we did before, computing the length of x,.(§S-) by using (3.12), we may put

68

(3.13) h'(s) = cos f(s), k'(s) = sin f(s)

for some smooth function f satisfying f(0) = 0.

On the other hand, we obtain from (3.12)
k"(0) = 0(8/85, 8/85) (0, 9)
= 0'( cos 9 e1+ sin 9 e2, cos 9 e1+ sin 9 e2)
= K(0) cos 29 N1 + K(O) sin 29 N2 .
Since k"(0) = (cos f(0)) f(0) = f(0) = i K(O),
(3.14) N(9) = i (cos 29 N1 + sin 29 N2).

Thus, for a suitable choice of Euclidean coordinates of E4 associated with e1, e2, N1 and

N2, the immersion x is locally determined by

S S S
(3.15) x(s, 9) = (cos 9 Jcos f(t) dt, sin 9 jcos f(t) dt, i cos 29 j sin f(t) dt,
0 0

S
i sin 29 j sin f(t) dt ),
O

S

where f(5) = i- J K(t) dt and K is the Frenet curvature of geodesics through 0.

If a surface M has the form (3.15), it is easily checked that M satisfies (*3).

69

Thus we conclude

Theorem 3.6 (cf. Theorem 3.8). Let M be a surface in E4 without isolated flat
points. Then M satisfies the property (*3) if and only if M is locally a surface of

revolution which lies in E3 or a surface that locally has the form (3.15).

Corollary 3.7. Let M be a complete connected surface without isolated flat
points in E4. Then M satisfies the property (*3) if and only if M is a surface of revolution

which lies in E3 or a surface is globally of the form (3.15) .

We now consider a surface M which lies in E5 satisfying the property (*3).
Again, we assume the base point in the property (* 3) is the origin of E5, where o is not an
isolated flat point.

If dim (Im (5)0 S 2, then M is locally a surface of revolution in E3 or M is a
surface with local representation about 0 of the form (3.15) which lies in E4 by the exact
same argument .

Suppose dim (Im (3)0 = 3. Then
0(61. 61) A 0(61, 62) A 0(62. 62) it 0.

Choose three orthonormal normal vectors to M at o :

(3.16) N1 2%), N2=9_(e;_,e_2), N3=—,I:LL,
K(O) ||N3||

where a = II 6(e1, e2) II and N3 = C(62, e2) - < 6(e1, e1), N1 > N1. If we compute the

second fundamental form at o as we did to derive (3. 14), then we obtain

70

K(0)2 -2b2

)N1+asin29N2+bsin29N3,
K(0)

6(8/85, 8/85) (0, 9) = ( K(O) cos2 9 -

where b = u i113 ||. Using this equation and (3.5), we can find N(9) :

2 _ 2
N(9) =: ( cos2 9 3“»—21) N1 i—a—sin 29 Nzi-P— sin2 9 N3.
K(0)2 K(O) K(O)

Thus locally the immersion x : M —-) E5 may be written in terms of a suitable choice of

Euclidean coordinates of E5 as
S S
(3.17) x(s, 9) = (cos 9 Jcos f(t) dt, sin 9 Ojcos f(t) dt,

K(0)2 -2b2 5

i ( cos2 9 - K(0)2 )(j sin f(t) (it,

a . s . b . s .
i—srn 29 srnft dt, i— 51n29 srnft dt ,
K(O) () K(0) J () )

S

where f(5) = i OJ K(t) (it and K is the Frenet curvature of geodesics through 0.

Conversely, if a surface M has the form (3.17), then we can easily see that M

satisfies the property (*3).

Theorem 3.8 (cf. Theorem 3.6). Let M be a surafce in E5 without isolated flat
points. Then M satisfies the property (*3) if and only if M is locally a surface of

revolution in E3 or a surface in E4 which has a local representation of the form (3.15) or a

71

surface of the form (3.17) which fully lies in E5.

Let M be a surface in E". Since the dimension of the first normal space at o is at

most three, we obtain the following theorem.

Theorem 3.9. Let M be a surface in Em (m 2 3) without isolated flat points.
Then M satisfies the property (*3) if and only if M lies locally in E5 and M is one of the
three model spaces described in Theorem 3.8.

§ 2. Surfaces of a Euclidean space with planar geodesics through
a point which is an isolated flat point

In this section we study a surface M in Em satisfying the property (*3). Base
point 0, say the origin of Em, is an isolated flat point.
We now assume that M is an analytic surface in E“.

We first define the degree of an isolated flat point.

Let p be an isolated flat point of a Riemannian manifold. Then for every

geodesic 7 parametrized by the arc length 5 through p = 7(0), its curvature K.Y(s) satisfies
KY(0) = 0 and Ky(s) :6 0 for sufficiently small 5. Let d(p) = inf { n e Z+ | KY(“)(0) :6 0 }.

Then d(p) is well-defined integer 2 1.
Definition. d(p) is called the degree of the isolated flat point p.

Suppose that the analytic surface M in Em satisfies the property (*3) and that the
base point in the property (*3) is an isolated flat point. We also assume that the base point

is the origin 0 of E".

According to (3.5) and Lemma 3.5, the immersion x : M —> Em is locally

72

represented in terms of geodesic polar coordinates (s, 9) about 0 :
(3.18) x(s, 9) = h(s) e(9) + k(s) N(9),

where h and k are analytic functions of s such that h(0) = k(O) = 0, h'(s) = cos f(s), k'(s)
5

= sin f(s), f(5) = :L-J K(t) dt , e(9) = cos 9 e1 + sin 9 e2 and N(9) is a unit vector normal

to M at 0 depending on 9.

For r 2 2, the r-th derivatives of h and k are :

h(‘>(s) = - (sin f(s)) 96%) + O1(f ', f 962))
and

k(’)(s) = (cos f(s)) 9"1)(s) + 02(f ', f 962) ),
where Oi(f ', f ",..., f(r‘z) ) (i = 1, 2) are certain polynomials with respect to f ', f ",...,
t5”). Since 0 is an isolated flat point and since the curvature of each geodesic through 0 is
independent of the choice of 9, there is an integer p ( > 1) such that K(O) = K'(0) = =
K(P-2)(0) = 0 and K(P'1)(0) at 0, that is, the degree of o is p -1. Since K(s) = i f '(s), we
see that

h<’)(0) = 0 for any r 2 2,

and

90(0) = 0 (o s r s p — 1), k(P)(0) ¢ 0.

73

In other words,

8rx

(3.19) —(0,9)=0 (2SrSp-1)
8sr
and
3px 9) (p-l)
(3.20) — (0, 9) = k (0) N(9) = K (0) N(9) as 0.

8sp
We now define the r—th (r 2 1) covariant derivative of o by

(V’cxxi. x2, Xr+2) = Dx1((\‘7"1c)< x2, Xr+2))

r+2

- 2 (7"ch x2, VXIXi, x42).
i=2

Then Va is a normal bundle valued tensor of type (0, r+2). Moreover, it can be proved

that
(3.21) (V’cxxi. x2. x3, x42) - (Y‘cxx2, xi. X3, x142)

= RN(X1, x2)( (Vr'zoxxr Xr+2))

r+2

+ 123: (Vr'ZGX X3, R(Xl, X2)Xi. Xr+2)

for r 2 2, where X1, X2, X3, ..., Xr+2 are vector fields tangent to M, RN the normal

74

curvature tensor, R the curvature tensor of M and V00 = o.

On the other hand, for r e Z+,

8’x

a—S;(o. e) = (V’cxw).

where t = e(9) = cos 9 e1 + sin 9 e2 and (Vro)(t‘+2) = (Vro)(t, t, t,...,t).

Lemma 3.10. If (Vr0)(t’+2) = 0 (0 s r s p -1) and (VPoxtPtZ) :6 0 for allt
in TM , then Vpo is symmetric and VIC = 0 at the point 0 for 0 S r S p -1.

Proof. For p = 1 , it is clear. Let p = 2. Since o(t, t) = 0 for every t e TOM, o
is a flat point of M. So the curvature tensor R vanishes at 0. Also, (Vo)(t3) = 0 for all
t e TOM implies VG = 0 at the point 0. Suppose it is true for p = r - l (r > 3). By the

induction assumption and (3.21), we obtain the result. (Q. E. D.)
Thus if the point 0 is an isolated flat point of degree p -1, then we have
Vro=0 (0SrSp-1)
and

Vpo is symmetric

at the point 0.

Denote by

75

_ i - _
(Va) (61. 65) = (V’6 )(611. 612..... en , 621. 622..... 621').

where i +j = r + 2 and em = e1, e2k = e2 for all h = 1, 2,..., i and k = 1, 2, 3., j. Since

the curvature K is independent of the choice of the geodesic through 0, we get

n (We) (cpfz) II = n (We) (e(9)P'1*2) II for all 9.

So we obtain the following equation.

+1
3: (9:2) cos 2W2") 9 sin 2’ 9 II (6%) (e172) "2
r=l

+1
-: (pi-2) cos 29’9") 9 sin 2’ 9 II (Vpo) (ep+12'r, e5) “2
r=l

+ 22(pi2) (p:2)cos 2(pJ'22'r'S9sin ”S 9 < (Vpo) (ep+2'r, e5), (Vpo) (e1212 ’8, e3) >

I'<S

which yields

(3.22) (P32) II (6%) (e972) "2

76

= (9:2) It (VPo) (ep+12*1, cg) “2

+2 2 (W2) (“2) < (VPo) (eP+2“, e2), (VPo) (e1”'12,S e2) >

r<sS2q
forr =2q (qZ 1) and
(3.23) 2 (W2) (“2) < (VPo) (e912, ’ e2), (VPo) (eP+12, 3 e92) > = 0
r<sS2q- l

forr=2q-1(q21).
On the other hand, the maximal dimension of { (Vpo)( e11, e12, ..., Cip+2 ) l 6ij =
e1 or e2 } is p +3.

From (3.20), we see that

(3.24) N(9) = 1 (Vpc) (stop?)

ll (VPo)(ep‘1‘2) 112

 

+2
= 1 (pl-2) cos pa" 9 sin I 9 -
ll (VPo) (ep‘l‘2) ”2 i=0

 

(VPO') (ep+12, r e2).

Thus, if 0 is an isolated flat point of degree p-l, then M is locally contained in at
most (p + 5)-dimensional Euclidean space Ep+5 and in this case the second fumdamental

form 0' satisfies (3.22) and (3.23) and the immersion x : M —-> Em becomes

77

x(s, 9) = h(s) cos 9 e1 + h(s) sin 9 e2 + k(s) N(9),

where h and k are analytic functions such that h(f)(0) = 0 for all r 2 2, k(r)(0) = 0 (0 S r S
p-l), k(P)(0) ¢ 0 and N (9) is given by (3.24).

Thus we have

Theorem 3.11. If a surface M in Em satisfies the property (*3) whose base
point , say the origin 0 of E“, is an isolated flat point of degree p - 1, then M locally lies in

at most (p + 5)-dimensiona1 Euclidean space Ep+5 about 0 and is of the form :
S S
(3.25) x(s, 9) = ( jcos f(t) dt ) (cos 9 e1 + sin 9 e2) + ( 1' sin f(t) dt ) N(9),
0 0

S
where N(9) is of the form (3.24), f(5) = i (J K(t) dt and K(s) is the Frenet curvature of

geodesics through 0.

Combining Theorem 3.9 and Theorem 3.11, we can classify analytic surfaces in

Em satisfying the property (*3).

Theorem 3.12 (Classification). Let M be an analytic surafce in E“. If M
satisfies the property (*3), then M is one of the following :

(1) M is locally a surface of revolution about 0 which lies in E3,

(2) M is a surface of the form (3.15) about 0 which fully lies in E4,

(3) M is a surface of the form (3.17) about 0 which fully lies in E4,

(4) M is a surface of the form (3.25) about o which lies in EP+5, where the

 

78

degree of the isolated flat point 0 is p -1.

Remark. According to K. Sakamoto, [ S-l ], and J. A. Little, [ L], a surface
in a Euclidean space Em with planar geodesics must be an open portion of a plane E2, a
standard sphere S2 or a real projective space RP2. So M must lie in a 5-dimensional
Euclidean space E5. However, a surface M in Em (m 2 3) satisfying the property (*3) may
lie fully in a higher dimensional Euclidean space depending on the degree of the isolated

flat point if the base point in the property (*3) is an isolated flat point.

CHAPTER 4

SURFACES IN A PSEUDO-EUCLIDEAN SPACE E':
WITH PLANAR NORMAL SECTIONS

§1. Some fundamental concepts and definitions

We introduce some basic terminologies and definitions for later use.

If In denotes the unit matrix of degree n, we put

Ipsq=(1(i) 1%

sl(n, R) = { all n x n — real matrices of trace 0 }.

Definition. ([ B-2 D. The yezenese immersiens of signature (r, n-r) are defined

{2(n+l) 1 1 n .. 2
X6 S:( T)CE?+ —) :- m (X X - 31114.1) 6 80(1‘, n+1-r),

:1:
where x = tx Ir,n+1_r , and

79

80

 

1 n 2
)c E2421") m (x *x + 31114.1 ) e so(r+1, n-r),

XEHr(

where a"x = tx 1H1,“ , and
50(p, q) = { A e sl(p+q, R) | IP, th Ip, q = A}.

In the first case, we take <A, B> = tr(AB) for the metric, so that 50(r, n+1-r) E Elia, +12)

with N: —n(n + 3). The image of S? is contained in S f(n +1 l_r)(1) with zero mean curvature.
In the second case we use < A, B > = - tr(AB) as the metric, so that so(r+1, n-r) '5

EN (r+1)(n- r) , and the Image of Hr has zero mean curvature in HN- ( r3131“ r)_ 1(1). Both

immersions are isometric and they are planar geodesic immersions.

Notation. E': t denotes Em with symmetric bilinear form < , > whose signature

has 5 negative, In - s -t positive and t zero signs, that is, t independent directions which
are orthogonal to everything. When t = 0, we have the pseudo-Euclidean space E21.

Definition. An isometric immersion i: MI1 -—> Em t is parallel if its second

fundamental form 0 is covariantly constant , that is, Vo = 0. An immersion of a pseudo-

Riemannian submanifold M? into Eng,t is full if its image is contained in no affine

hyperplane of E? t
. . n .
For any funct1on g . Er —> R, the mapping

n+2
r+1

x e E’; l> (s(x). x. s(x» e E

81

is a planar geodesic immersion with < 0'(X, X), c(X, X) > = 0 for all unit vectors

1
x e T132, which is full in ET] (if g is not linear). This immersion is called an expansion
n . n+1
of ISI 1nto E121.

If the immersion i is parallel, then the geodesics are mapped to parabolas or line
segments, so the function g associated with expansion must be a quadratic polynomial. Up

1 n
to isometry of E221 , g = Zai x12, so that i(E':) is an elliptic or hyperbolic paraboloid or an
' i=1

orthogonal cylinder over one of these .
Let g1, ..., gk be independent quadratic polynomials from E? to R, k S %n(n+1).

n+k
Then we can define an expansion E? into E rk by

i n+k
x 6 E: -> (510‘). $3200, gk(X), X. g1(X). g2(X). gk(X)) E E r'k-

Let M? be a pseudo-Riemannian manifold which lies in a pseudo-Euclidean space

B? We define a normal section of M? at a point p e M? in a direction t (at 0) e TpM'rl in

a similar way that we did in Chapter 1 : Let E(p ; t) be an affine space spanned by t and
'1ng . Then the intersection of Mr: and E(p ; t) gives rise to a curve in a neighborhood of

p which is called the normal section of M: at p in the direction t. M? is said to have planar

nermal seetieng if every normal section 7 is a plane curve, that is, 7A 7" A 7" = 0. MI; is
said to have peintwise planar nermal seexieas if 7(p)A 7'(p) A 7"(p) = 0 for every point

D
peMr.

§ 2. Surfaces in a pseudo-Euclidean space E': with

planar normal sections

82

We now suppose that a (pseudo-Riemannian) surface M3 in a pseudo-Euclidean

space E]: has planar normal sections. Let M = M? (r = 0, 1, 2) in order to make the matter

simple.
Let p be a point of M and let t be a nonzero vector tangent to M at a point p and let 7

be the normal section of M at p in the direction t. We assume 7(0) = p. Let T = 7 '(s),

where sis a parameter (which is not necessarily the arc length ). Then we have

(4.1) 7 "(s) = VTT + o(T, T),

(4.2) 'Y "'(S) = VTVTT + C(VTT, T) - ACCT, T)T + DT C(T, T).

7 "'(0) is a linear combination of 7(0) and 7'(0) at 7(0) = p because 7 is planar.

So we have
(4.3) VtT = at for some a e R,
(4.4) (VtGXt, t) A o(t, t) = 0,

where t = T(0). But (4.4) is true for any point p and any nonzero vector t. Thus we have

(4.5) (VXcXX. X) A c(x, X) = o

for every X 6 TM. In particular, if t is nonnull, that is, < t, t > at 0, then we can reduce

(4.3) to

83
(4.6) VtT = 0.

Proposition 4.1. Let M be a surface in a pseudo-Euclidean space E21. Let 7 be

a planar normal section of M at p in the direction of a unit vector t. If 7"(0) is null, then
o(t, t) is a null vector.
Proof. Since 7 is a plane curve which lies in the plane spanned by t and 0(t, t),

we have
7(5) = 7(0) + f(s)t + g(s)o(t, t)
for some function f and g along 7. This implies
7"(5) = f'(s)t + g"(s)0'(t, t).
On the other hand, we see that
Y'(0) = F'(0)t + g"(0) 0(t. t) = 0(t. 0

because of (4.6). Thus, f"(0) = 0 and g"(0) = 1. Since 7"(0) is null, 0(t, t) is a null
vector. (Q. E. D.).

Let t be any nonzero tangent vector of M at p and let 7 be the normal section of M at
p in the direction t. Since 7'A 7' A 7'" = 0, 7'" is a linear combination of 7' and 7".

(4.1) and (4.2) imply

(4.7) VTVTT - Ao(T, T)T = b(s) T + c(s) VTT ,

84

(4.8) o(VTT, T) + DT o(T, T) = c(s) o(T, T)

for some functions b(s) and c(s).

Suppose that 7 is a geodesic. Then for an affine parameter 5,

VTT = 0.

Hence we have
YKS) = C(T, T), Y"(S) = ' AO(T, T)T '1' DT C(T, T).

Since 7'A 7' A 7'" = 0, we obtain

(4.9) AG(T, T)T A T = O,
that is,
< o(T, T), o(T, Ti) > = 0

for all Ti 6 TM such that < T, Ti > = 0.

Suppose that 7 is not a geodesic. Then VTT #5 0 for s at 0. Since (VTo)(T, T) A

0(T, T) = 0, we get from (4.8)

(4.10) 0(VTT, T) A 6(T, T) = 0.

In particular, if t ( ¢ 0) is not null, then by the arc length parametrization, we may

85

assume
<Y®1Wfi>=e. 8=iL

Choose an orthonormal frame [ e1, e2} on a neighborhood U of p such that e1 is an
extension of T = 7(5). Then VTT = p(s) e2 for some function p since < VTT, T > = 0.

By using continuity, we have

Lemma 4.2. Let M be a surface of a pseudo-Euclidean space E? with planar

normal sections. If the normal section 70f M at p in the direction of nonnull vector t is not

a geodesic on a sufficiently small neighborhood, then we have
canaoae)=0

where t = 7(0) and ti is a unit vector in TM perpendicular to t.

Lemma 4.3. Let M be a surface in a pseudo-Euclidean space E': with planar

normal sections. If the normal section 7 is a geodesic are on a neighborhood of p, then we

have
<oan,oat5>=0

where t = 7(0) and ti is a unit vector in TpM perpendicular to t. ( In this case, ti may be t

if t is a null vector.)

If M has planar normal sections, then M clearly has pointwise planar normal

sections. We now prove

86

Lemma 4.4 ([ Ch.B-S ]). Let M: be an n-dimensional pseudo-Riemannian

submanifold of a pseudo-Euclidean space E? If M: has pointwise planar normal

sections, then Im a is parallel in the normal bundle.

Proof. As we derived (4.4), we can see that M? has pointwise planar normal

sections if and only if

(4.11) (VXo)(x, X) A c(X, X) = 0

for any X 6 TM?. For any two vectors Y, Z 6 TM?, if we set X = Y + Z, then (4.11)

gives

(4.12) (VYo)(Y, Z) + (VZo)(z, Y) 6 Im 0'.

Similarly, if we set X = Y - Z, we have
(4.13) (VYo)(Y, Z) — (VZo)(z, Y) 6 Im o.
Combining (4. 12) and (4.13), we obtain

(VYo)(Y, Z) 6 Im o.

Together with the Codazzi equation, this implies

(on)(Y, Z) 6 Im o.

87

for any X, Y, Z 6 TM? Thus we conclude that the first normal Space 1m 0' is parallel in

the normal bundle. (Q. E. D.)

Using this lemma and a reduction theorem , [ CH.B-5 ], [ E], we have

Proposition 4.5. Let M? be a pseudo-Riemannian submanifold of E? with
dim (Im 0') = k for some integer k. If M: has pointwise planar normal sections, then M:

lies fully in an affine space of dimension (n+k) c: E21.

We now talk about the dimension of the first normal space of a surface in E':.

Proposition 4.6. Let M be a surface in a pseudo-Euclidean space E? with index

r (r = 0, 1, 2). Suppose that
< «X. X). c(X. Xi) > at 0

for any orthonormal basis { X, Xi } of TpM, p e M. Then dim (Im 6);, S 1.

Proof. Let {e1, e2} be an orthonormal basis of TpM such that
< 0(61, 61), 6(61, 62) > at 0.

Suppose that r = 0 or 2, that is, M is spacelike or timelike. By continuity, there is

an open interval (a, b) of 9 such that

< o(e(9), e(9)), o(e(9), c(9)i) > 4 0

88
for any 9 e (a, b), where
(e(9) )= (cos 9 sin 9)(el
e(e)J- — sin 9 cos 9 e2
By Lemma 4.2 and Lemma 4.3 we obtain
0'(e(9), e(9)) A o(e(9), e(9)J') = 0
for all 9 e (a, b). A straightforward computation shows that
6(61. 61) A 6(61. 62) = 6(61. 61) A 6(62. 62) = 6(61. 62) A 6(62. 62) = 0.

Suppose that r = 1, that is, M is Lorentzian.

Let
e(9) _ cosh 9 sinh 9 (e1
(3(6)i sinh 9 cosh 9 62
By a similar argument to that above, we get

6(61, 61) A 6(61. 62) = 6(61, 61) A 6(62. 62) = 6(61. 62) A 6(62. 62) = 0.

Therefore we conclude that dim (Im (3)1, S 1. (Q. E. D.)

Lemma 4.7. Let M be a surface of a pseudo-Euclidean space Eng. Then

89

< 0(X, X), 0(X, X) > does not depend on the choice of the unit vector X 6 TM if and
only if < O'(X, X), (s(X, XJ') > = 0 for any orthonormal basis { X, XJ' } of TM.

Proof. The proof is by the same argument as in Lemma 1.1. (Q. E. D.)

Definition. Let Mr; be a pseudo-Riemannian submanifold of a pseudo-

Riemannian manifold M11. M? is said to be We 6 M: if the length of the
second fundamental form does not depend on the choice of unit vector in TpM‘r'. M? is said

to be pm if M? is pseudo—isotropic at every point of M3. If M? is pseudo-

isotropic and the length of the second fundamental form does not depend on the choice of
the point, then we say M: is constant pseudo-isotropic,

Lemma 4.8. Let M be a surface in a pseudo-Euclidean space E? with index r (r =

0, 1, 2). If M is pseudo-isotropic at p, then we have
(4.14) (-1)r < 0(61. 61) . 6(61. 61) > = <0(61. 91), 6(62. 62) > + 2 < 0(61. 92), 0(61. 62) >,

where [ e1, e2} is an orthonormal basis for TpM.
Proof. Let { e1, e2} be an orthonormal basis for TpM and let { X, Xi } be
another orthonormal basis for TpM.

Suppose that r = 0 or 2. Then there exists 9 e [ 0, 2n ) such that
X=cos9e1+sin9e2, Xi=-sin9e1+cos9e2.

Substitution of X and Xi into < c(x, X), o(X, Xi) > = 0 gives (4.14).

Suppose that r = 1. We put

90

X = cosh 9 C1 + sinh 9 e2, XJ' =sinh 9 e1 + cosh 9 e2

for some 9. Substitution of X and Xi into < o(X, X), c(X, Xi) > = 0 gives (4.14).

(Q. E. D.)

C. Blomstrom, [ B-2 ], generalized the notion of planar geodesic immersions in the
Riemannian case to that of the pseodo-Riemannian case and she proved that ifi : M? —> Br:

. . . . n . . . .
15 a planar geodesrc rmmersron, then Mr 15 constant pseudo-isotroprc. From now on, if

M? is constant pseudo-isotropic, then we denote < 0(X, X), 0(X, X) > by L, where X is

a unit vector tangent to M:

Lemma 4.9 ([ B-2 ]). Let i : MI; —) E? be a planar geodesic immersion with
L = 0. Then M? is flat if and only if the first normal space Im o is composed entirely of
null vectors. In particular, if M? is complete, then i must be (up to rigid motion) an

k
expansion of E: into E221} , where k = dim(Im 6).

Proposition 4.10. Let M be a surface which is constant pseudo-isotropic in

4

E r +1 with L = 0. If M has planar normal sections, then Im o is a 1-dimensional null

space, that is, M lies in E3 1.

1
Proof. Let p e M. Then TpM is isomorphic to E2. Let { e1, e2 } be an

orthonormal basis for TpM. Then we have
< 0(61, 61). 6(61. 62) > = 0.

Suppose 6(e1, e1) A o(e2, e2) :6 0. If O'(61, e2) = 0, by (4.14) and the assumption

91

L = 0 we obtain
< 6(61. 61). 6(62. 62) > = 0.

Thus , Span { o(e1, e1), 0(e2, e2) } is isomorphic to E2) '2. But this is impossible in T;M

E E2. Thus we get 6(e1, e2) 96 0. Since < 0(e1, e1), 0(e1, e2) > = 0 and since T;M is a
Lorentzian vector space,
6(61. 61) A 6(61. c2) = 0 and 6(62. 62) A 6(61. 62) = 0.

In this case, we also have Span { 0(e1, e1), 6(62, e2) } is isomorphic to E2 2. Hence we

have
6(61. 61) A 6(62, 62) = 6(61. 62) A 6(61. 61) = 6(61. 62) A 6(62. 62) = 0.

Consequently, Irn o is a 1-dimensional null space at every point of M. By Lemma 4.4,
Irn o is parallel in the normal bundle. Thus M locally lies in E2 '1. (Q. E. D.)

Remark. Under the same assumptions as in Proposition 4.10, M is flat by the
similar proof of Lemma 4.9. Thus, if M is complete, then i : M —> E"; must be an

expansion of E2 into E2 1.

Blomstrom, [ B-2 ], proved

Proposition 4.11. If i : M —> E: is parallel and full, then

92

«Km. 61). 0(62. 62) > - 2 < 0(61. 62). 0(61. 62) > = 0.
where { e1, e2 } is an orthonormal basis of TM.

Proposition 4.12. Let M be a constant pseudo-isotropic Lorentzian surface with

L = 0. Then
<o(t, t), c(t, t) > = 0

for all null vector t e TpM.

Proof. Since M is pseudo-isotropic,
< o(X, X), o(X, xi) > = 0
for any orthonormal basis { X, Xl } of TM. Let t be a null vector of TpM. Then
t=aX+aXi
for some a ( ¢ 0) e R. By using (4.14), we have
<o(t, t), 0(t, t) > = 2a2 { < c(x, X), c(xi, xi) > + 2 < o(X, xi), o(X, xi) > } = 0.
(Q. E. D.)

Proposition 4.13. Let M be a constant pseudo-isotropic surface of a pseudo-
Euclidean space 13': with index 'r (r = 0, 1, 2) which has planar normal sections. If L =0,

93

then M has planar geodesics.

Proof. Suppose r =1. Let t 1: 0 be a nonnull vector at p e M and let 7 be the
normal section of M at p = 7(0) in the direction t. Since < t, t > :t O, we may assume that y
is parametrized by the arc length s. So < T(s), T(s) > = e for sufficiently small 8, where
‘y'(s) = T(s) and Y(O) = t.

Suppose o(t, t) ¢ 0. Then there exists a neighborhood U of p such that o(u, u) ¢ 0
for any unit vector u e T(U). Since 7 is a plane curve which lies in a plane spanned by t

and o(t, t), we may express

(4.15) 7(3) = p + f(s)t + g(s)o(t, t)

for some functions f and g defined on an interval 1. Then we get

”((8) = f(S) t + g'(S)G(t. t).

7"(5) = f"(S) t + g"(s) o(t, t).

Since < Y(s), Y(s) > = e and since L = O, we get

(f'(s))2 = 1.

Thus the last equation becomes

Y'(s) = g"(s) 6(t. t).

Therefore, < Y'(s), Y'(s) > = 0 on I.
On the other hand, Y'(s) = VTT + o(T, T) and thus < VTT, VTT > = 0 because of

94

L = 0. Choose an orthonormal frame [ T, T‘L ] along 7. Then VTT A TJ‘ = O and hence

VTT = O on I. Thus 7 is a geodesic arc.
Suppose 6(T, T) = O for some interval containing 0 , where 7(0) = p. Then 7 is

clearly a geodesic arc.
Suppose o(t, t) = O and 0(T, T) at 0 for s > 0. Since L = 0, o(T, T) is null for
s > O. Extend 0(T, T) up to p = 7(0) and denote it by n. Then

7(8) = p + f1(S)t + gl(S) n
for some functions f 1 and g1. Using an argument similar to the one developed above, we
see that y is a geodesic arc.
Let t be a null vector tangent to M at p. Let 'y be the normal section of M at p in the
direction t and let 7(0) = p and 7(0) = t. By Proposition 4.12, we have
< o(t, t), o(t, t) > = 0.
We may put
7(8) = p + f(s)t + 23(8) 60. t)
for some parameter s and some functions f and g. Let 7(3) = T. Then , we get
T = f'(s)t + g'(s) 0(t, t)
and

VTT + c(T, T) = f'(s)t + g"(s) o(t, t).

95

Since < t, t > = < o(t, t), o(t, t) > = 0, we have < T, T > = 0 and hence

< o(T, T), 0(T, T) > = 0 by means of Proposition 4.12. Thus we obtain

Suppose there is so such that T(so) A VT(so)T at 0. Then T(so) and VT(so)T form a
degenerate plane E33. But this is impossible because TY(SO)M .-':- Ei Thus we have

So VTT = h(s) T for some function h. 7 is indeed a pregeodesic, that is, it has a

parametrization as a geodesic. Thus, 7 becomes a geodesic 7 by changing its parameter

such that 7 (s) = (yoaXs) satisfying a" + h(a')2 = 0.

We now suppose that the surface M is spacelike or timelike, that is, r = O or 2.

Let p be a point of M and 'y be a normal section of M at p in the direction t ( ¢ 0).
Using the exact same argument as that of the first half of the case r = 1, we see that y is a
geodesic arc.

For a given nonzero vector t, the geodesic with initial velocity vector t is unique and

hence it is the normal section. Therefore, M has planar geodesics. (Q. E. D.)

Lemma 4.14. Let M be a constant pseudo-isotrOpic surface in a pseudo-
Euclidean space E': with L = 0. If M has planar normal sections and if the mean curvature

vector H is parallel in the normal bundle, then Im o is spanned by null vectors.

Proof. Let [ e1, e2 } be an orthonormal basis of TpM, p e M. Then the mean

curvature vector H is given by

96

1
=-2-{ <61,61>O'(61,61)+<62.62>0(32s€2) }:

which implies

<H, H>=%(-1)r<o(e1,e1),o(e2,e2) >.

Suppose that < H, H > at 0. Extend e1 along the geodesic 71(3) = expp(s e1) which
is denoted by E1 and extend e2 by parallel displacement along 71 which is denoted by E2.
By Lemma 4.8 and L = O, we get

< 0( E1, E1 ). 0( E2. E2) > =(-1)”1<H.H >.
Since the mean curvature vector is parallel in the normal bundle, we get

131(1)) < 0( E1. E2 ). 6( El, E2) > = 0-

Since VElEl = 0, we also get VE1E2 = 0. Thus we obtain
(4.16) < (Velox e1. e2 >, o< e1, e2) > = 0.

On the other hand,
(4.17) O = E1(p) < H, H >

= E1(1)) < 0( El, E1 ), 0( E2, E2) >

97
= < (Velox e1. e1 ). o< e2. as) >
+ < G(61.61). (velox 62.62) >.

Extend e2 along the geodesic 72(3) = expp(s e2) and denote by E2 and extend e1 to

E1 by parallel displacement along 72. Then we get
(4.18) 0=E2<p) <o<E1.E1).o(E1.E2>>

= < (Vezox 61. 61 ). G( 61. 62) >
+ < 0( 61. 61 ). (Vezox 61. 62) >.

Combining (4.16), (4.17) and (4.18) and making use of (4.5), we obtain

(\‘7610><e1.e1>= 0.

Since el can be chosen arbitrarily, we have

Vo=Q

that is, the immersion i : M —> E? is parallel.
Since dim (Im o) S 3, i : M —> E1: is full in at most a 5-dimesional subspace of E?

by Lemma 4.4. However, Proposition 4.10 implies that i cannot be full in E: +1 . If i is full

98

in EE, < o( e1,e1 ), o( e2,e2) > = O by Lemma 4.8 and Proposition 4.11. This contradicts

<H,H>¢O.If dim(Imo)=1, 0(e1,e1)Ao(e1,e2)=O, 0(e1,e1)Ao(e2,e2)=O

and 0( e1, e2) A 0( e2, e2) = O . Thus Im o is a l-dimensional null space. So, i cannot
be full in E2. Thus i is necessarily full in a degenerate 3, 4 or 5-dimensional affine space of

E2}. Therefore, the scalar product of one of 0( e1, e1 ), 0( e1, e2) and 0( e2, e2) with any

vector vanishes. By considering Lemma 4.8, we get
<0(61.61). 0(62.62)>=0.

Consequently, < H, H > = 0 at p. Since p is arbitrary, < H, H > vanishes identically on

M. From this and Lemma 4.8, we conclude that Im o is spanned by null vectors.

(Q. E. D.)

Theorem 4.15. Let M be a constant pseudo-isotropic surface in E? with planar

normal sections and L = 0. If the mean curvature vector field H is parallel in the normal
bundle, then M is flat. Moreover, if M is complete, then i : M —> E1: is an expansion of

+k
133 which is full in 133k (k = 1, 2, 3).

Proof. By Lemma 4.14, < H, H > = 0 and Im o is generated by null vectors. We

see that M is flat by considering the Gauss equation (0.7). So, if M is complete, then
k .
i : M —> 13‘: is an expansion of 133 into 1331,", (k = 1, 2, 3) by Lemma 4.9. (Q. E. D.)

We now prove

Lemma 4.16. Let M? be an n-dimensional pseudo-Riemannian submanifold in a

pseudo-Euclidean space E? with pointwise planar normal sections. If Mr; is pseudo-

99
isotropic, then M? is constant pseudo-isotropic.

Proof. Let < 0(X, X), c(X, X) > = f(p) for unit vectors X e TpMIr1 . Then f is a

function on M? . We shall prove that f is constant.

Choose a point p e M? and a geodesic “y emanating from p with nonnull initial

velocity vector t. We assume that 'y is parametrized by the arc length 3. Let tl be a unit
vector in Tper1 such that < t, ti > = O. Extend t‘L to a vector field Ti tangent to M?

which is parallel along "y and extend t = 7(0) on a neighborhood of p = 7(0) such that T =
Y(s) stays a unit vector field orthogonal to TJ‘. Since M? is pseudo-isotropic and M has

pointwise planar normal sections, we get
0 = t < 60. T). o<T. Ti) >

= < Dto(T, T), 0(t, 1*) > + < c(t, t), Dto(T, Ti) >

= < 6(t, t), Dto(T, Ti) > (because Dto(T, T) A 6(t, 1) = 0)
= < o(t, t), (fitoXt, ti) > (because VtTi = O)

= < c(t, t), (VtioXt, t) >

1* < o(T, T), o(T, T) >

NIH

= %- tJ‘(f).

Since dim M? 2 2, f is a constant. So, Mr; is constant pseudo-isotropic. (Q. E. D.)

100

Theorem 4.17. (Classification). Let M be a surface in a pseudo-Euclidean space
E? with planar normal sections. If M does not lie in a 3-dimensional affine space of E?

and if the mean curvature vector field H is parallel in the normal bundle, then M is an open
+
portion of either a flat surface which locally lies in Ei'kk (k = 2, 3) or a Veronese surface

orE5

. 5
1“ E 5-(r+l)(2-r)'

r(3-r)

Proof. Let M1 = { p e M l dim (Im 0') S 1 at p }. Suppose M = M1. Since
Irn o is parallel in the normal bundle, we see that M lies in a 3-dimensional affine space of
E? . If M - M1 ¢ Q, then M - M1 is an open subset of M. Let U be a component of
M - M1. Proposition 4.6 implies that M is pseudo-isotr0pic. By Lemma 4.16, we see that
U is constant pseudo-isotropic. Suppose L = O on U. Then Proposition 4.13 tells us that

U has planar geodesics. Lemma 4.14 and the Gauss equation (0.7) imply that U is flat and
+k
fully lies in Ei‘k (k = 2, 3) . Suppose L at 0. Since U has planar normal

sections, (fionX, X) A C(X, X) = O for X e TU. Since L is constant on U,
(vonX, X) = 0 for all X e TU. Thus the second fundamental form C is parallel in the

normal bundle, that is, 76 = 0. Let { e1, e2 } be an orthonormal basis of TpU for some
p e U. If 6(e1, e2) = 0, then Im 0' is 1-dimensional at p by Lemma 4.8. Therefore,
o(e1, e2) at O for any orthonormal frame [ e1, e2 } on U. We also have
o(e1, e1) A o(e1, e2) at O. In fact, if o(e1, e1) A 0(e1, e2) = 0, then o(e1, e1) is a null
vector. This contradicts L 1: 0. Making use of Lemma 4.2 and Lemma 4.3, we can

conclude that U has planar geodesic normal sections. By the classification Theorem of
Blomstrom, [B-2 ], U is an open portion of a Veronese surface in E234) or E3-“ +1)(2_r).

By continuity of L, we conclude that U = M. This completes the proof. (Q. E. D.)

SUMMARY

Let M be a surface of a Euclidean space E“. We define the property (*1), (*2) and
(*3) as follows :

(*1) : There is a point p in M such that every geodesic through p is a helix of the
same curvatures.

(*2) : There is a point p in M such that every geodesic through p is a normal section
of M at p.

(*3) : There is a point p in M such that every geodesic through p is planar.

Then we obtain the following results.
(1) Let M be a complete connected surface in E3. Then M satisfies the property (*1) if and
only if M is a standard sphere or a plane.
(2) Let M be a compact connected surface in E4. Then M satisfies the property (*1) if and
only if M is a standard sphere which lies in E3 or a Blaschke surface at a point which lies in
E4 of the form (1.15) and is diffeomorphic to RP2.
(3) Let M be a compact connected surface in E5. Then M satisfies the property (*1) if and
only if M is a standard sphere in E3 or a Blaschke surface at a point of the form ( 1.37)
which lies in E4 or a Blaschke surface at a point of the form (1.40) which lies in E5 or a
Blaschke surface at a point of the form (1.32) which lies in E5. All such pointed Blaschke
surfaces are diffeomorphic to RP2.
(4) Let M be a compact connected surface in Em (m 2 5). Then M satisfies the property
(*1) and (*3) if and only if M lies in E5 and M is one of four model spaces stated in (3).
(5) Let M be a compact connected surface in E5. Then M is a Veronese surface if and only

if M has a constant Gaussian curvature and satisfies the property (*1) and the base point is

101

102

not umbilical.
(6) Let M be a surface in E3. Then M satisfies the property (*2) if and only if M is locally a
surface of revolution.
(7) Let M be a complete connected surface in E3. Then M satisfies the property (*2) if and
only if M is a surface of revolution.
(8) Let M be a surface in E3. Then the property (*1) is equivalent to (*3).
(9) Let M be a surface in E4 without isolated flat points in E4. Then M satisfies the
property (* 3) if and only if M is locally a surface of revolution which lies in E3 or a surface
that locally has the form (3.15).
(10) Let M be a surface in E5 without isolated flat points. Then M satisfies the property
(*3) if and only if M is locally a surface of revolution in E3 or a surface in E4 which has the
form (3.15) or a surface of the form (3.17) which fully lies in E5.
(11) Let M be a surface in Em (m 2 3) without isolated flat points. Then M satisfies the
property (*3) if and only if M lies locally in E5 and M is one of three model spaces stated
in (10).
(12) If an analytic surface M in Em satisfies the property (*3) whose base point is an
isolated flat point of degree p-1(p > 1), then M is locally of the form (3.25) and M lies in a
linear subspace of dimension 3 p+5.
(13) Let M be an analytic surface in 13'“. If M satisfies the property (*3), then M is one of
the following :

(a) M is locally a surface of revolution in E3,

(b) M is a surface of the form (3.15) in E4,

(c) M is a surface of the form (3.17) in E4,

(d) M is a surface of the form (3.25) in Ep+5. In this case, the base point of the

property (*3) is an isolated flat point of degree p -1.
(14) Let M be a surface which is constant pseudo-isotropic in E: +1 with L = 0. If M has

103

planar normal sections, then Im o is a l-dimensional null space, that is, M lies in E3 1'

(15) Let M be a constant pseudo-isotropic surface of a pseudo-Euclidean space E? with

index r ( r = 0, 1, 2) which has planar normal sections. If L =0, then M has planar

geodesics.
(16) Let M be a surface in a pseudo-Euclidean space E': with planar normal sections. If

M does not lie in a 3-dimensional affine space of E? and if the mean curvature vector field

H is parallel in the normal bundle, then M is an open portion of either a flat surface which

+k
locally lies in Ef'k (k = 2, 3) or a Veronese surface in EEO-r) or E2“ +1)(2-r)°

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