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THE SPECTRUM OF SASAKIAN MANIFOLDS

BY

Yhuji Shibuya

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1979

ABSTRACT

THE SPECTRUM OF SASAKIAN MANIFOLDS

BY
Yhuji Shibuya

The spectrum of a manifold, which is the set of
eigenvalues of the Laplacian, is in some sense related to
the "pure tone" of the manifold.

There is an old question asking, "can you hear the
shape of the drum?", that is, to what extend can you
determine some geometric character of the manifold by
knowing its spectrum?

In particular, we are interested in the question of
whether a manifold is isometric to a sphere, if the spectrum
of the manifold is the same as that of the sphere. This
question has been affirmatively answered in the 1,2,3,4,5
and 6 dimensional cases. But it is an Open question for
other dimensions.

In this paper, we affirmatively answer this question
in the 5,7,9,ll and 13 dimensional cases under the assump-
tion that the manifold is a Sasakian manifold, which is a

contact manifold with certain integrability condition.

Yhuji Shibuya

For the proof, we first establish several curvature
properties of a Sasakian manifold, then study some geometric
implications of the vanishing of the contact Bochner curva-
ture tensor, and finally we use the asymptotic expansion
of the fundamental solution of the real heat equation to
express the spectral condition in terms of curvatures. The
main theorem is then obtained for the wider class of

spaces in which spheres are included.

ACKNOWLEDGMENTS

The author wishes to express his hearty thanks to
Professor David E. Blair who gave him insight in the
theory of contact manifolds, Professor Bang-yen Chen who
introduced him to the interesting theory of the spectrum
of Riemannian manifolds, Professor Gerald D. Ludden who
offered him a chance to study at Michigan State
University, and to Professors Shigeru Ishihara and
Kentaro Yano at Tokyo Institute of Technology for

teaching him differential geometry in general.

ii

TABLE OF CONTENTS

INTRODUCTION
CHAPTER I: SASAKIAN MANIFOLDS
§l. Contact Manifolds
§2. Almost Contact Manifolds
§3. Normal Almost Contact Manifolds
§4. Sasakian Manifolds

CHAPTER II: CURVATURE PROPERTIES OF SASAKIAN MANIFOLDS

§1-
§2-

§3.

§4.
§5.

§6.

Curvature of Riemannian Manifolds
Curvature Properties of K-contact Manifolds

Basic Curvature Pr0perties of Sasakian
Manifolds

n-parallel Ricci Tensor

dysectional Curvature

n-Einstein Manifolds

CHAPTER III: CONTACT BOCHNER CURVATURE TENSOR

§1.
§2.
§3.

§4.

§5.

Weyl Conformal Curvature Tensor
Bochner Curvature Tensor
Contact Bochner Curvature Tensor

Vanishing Contact Bochner Curvature
Tensor

Length of Contact Bochner Curvature
Tensor

iii

Page

10
15

17

21
21

26

27
29
35

37

39
39
4O

41

43

46

Page

CHAPTER IV: SPECTRUM OF RIEMANNIAN MANIFOLDS 51
§l. Laplacian 51
§2. Spectrum of Riemannian Manifolds 52
§3. Asymptotic Expansion 56

§4. Geometric Application of Asymptotic
Expansion 58
CHAPTER V: SPECTRUM OF SASAKIAN MANIFOLDS 63
APPENDIX: Spectrum of p-Forms of Sasakian Manifolds 74

BIBLIOGRAPHY 76

iv

INTRODUCTION

Let Q C E31 be a bounded domain with smooth boundary

an. The vibration of O are the functions

F:QX]R-9]R

with
2

(0.1) AF + a__g_ = 0
at

and

(0.2) Flamm = 0,
n 2

where A = - Z) -§3'.
i=1 axi

In order to study the vibrations F of Q,

necessary to study those F's of type
(0.3) F(x,t) = f(x)e(t).

with f:Q-*]R

8:]R41R,

because of the Stone-Weierstrass theorem.
For those F's, (0.1) is

Af

_.. _ .91.:
(0.4) f - e A,

it is only

where A has to be a constant. This I is connected to

the frequencies of our vibrations since
9” +>.e = 0.

This is one of the reasons why we are interested in

the spectrum of O, which is

Specm) = {0 < x1 3X2 3... },

consisting of all X's such that there exists an f with
f y'o, Af = lf, flaQ=O' Each l is written in Spec(Q)
a number of times equal to its multiplicity, that is,
dimif':Af = If}.

There are two main questions:

(1) Given 0, determine Spec(0).

(2) Does Spec(Q) = Spec(fl') imply that Q is

congruent to 0'?

By defining the spectrum of a compact manifold M, we
can also ask the similar questions. (See Chapter IV)

Inspite of the simplicity of the questions and its
physical background, the answers to these questions are
almost unknown.

As for the question (1) in the general case, there are
some results to estimate the smallest eigenvalue ll. ll
is also evaluated theoretically. But ll is not known
for even a general triangle or an ellipse, etc. Spec(M)
is known only for tori, spheres, projective Spaces and a
few other manifolds.

The question (2) is sometimes referred as a drum

problem. Let Q and Q' be two vibrating membranes in

3-dimensional Euclidean space. As we have seen Spec(Q)
represents the frequencies of the vibration, that is,

the pure tones. The question (2) means that if you know
What 0 looks like, can you determine the shape of 0'
under the assumption of Spec(0) = Spec(Q')? Since

f2c21R3 is a strange shaped tambourine or drum, we usually
say, "Can one hear the shape of a drum?" (See Kac [17])
The answer to this question in the general case is no.
Actually there is a counter example in a l6-dimensional
torus. (See Chapter IV) But there may be some h0pe if

the shape of the manifold M. is particularly nice. One

of the questions in this direction is whether the asser-
tion. Zhn: Spec(M9,g) = Spec(Sn,gO) implies that (Mp,g)
is isometric to (Sn,go) or not, where (Sn,go) is an
n-dimensional sphere with the standard metric go. I:
22,23,2

is an Open question to decide if’ Zhn is or is not valid

1 I
4 . 25 and 26 are known to be true. But it

for any n.

In this research we give an affirmative answer to

(a .

(ZS ), Z7 , 29 , “11 and 2313 under the assumption that
M? is a Sasakian manifold, which is a contact manifold
with some integrability condition.

The main tool used here is the asymptotic expansion

-lit

of Zie , where *1 6 Spec(M9,g).

In Chapter I, we review the properties of contact

manifolds and the definition of Sasakian manifolds mainly

following Prof. Blair's book, "Contact Manifolds in

Riemannian Geometry".

In Chapter II, after some review of curvature
tensors of Riemannian manifolds, we establish several
interesting curvature properties of Sasakian manifolds,
since the asymptotic expansion of Z}e-xit is expressed
in terms of curvature tensors.

In Chapter III, we compute the length of the so-
called contact Bochner curvature tensor, and establish
some interesting pr0perties involving the contact Bochner
curvature tensor.

In Chapter IV, we review the theory of the spectrum

of Riemannian manifolds following, "Le Spectre d'une

Variété Riemannienne," by M. Berger, P. Gauduchon and

. -X.t
1
E. Mazet, in which the asymptotic expansion of Z3e

is explained.

In Chapter V, by using all the tools prepared, the
main theorem is proved.

Theorem. ‘Lgt. (M,g) Egg. (M’,g’) be compact
Sasakian manifolds with structure tensors (¢,§,n) and
(m',§’,n') respectively. Assume Spec(M,g) = Spec(M’,g’),
then we have

(1) dim M = dim M' {known},

(2) £9; dimM=dim m’=5,7,9,11, M M

constant m-sectionalicurvature c, if and
only if M’ is of constant m-sectional

I

curvature c = c ,

(3) for dim M.= dim M7 = 13, M is of
. 197
constant m-sectional curvature c # 7:;-,
if and only if M' is of constant m-
sectional curvature c = c .
As a corollary we see that (Z: ), Z) , Z} , Z} and
5 7 9 11
2313 are true under the Sasakian assumption on Mn.
The method used here is motivated by Tanno, who proved
Z4 , 25 and 26 without any assumption on Mn.
Einstein's summention convention is used for tensor
calculus. The notations Z : integers, R: the real number

field, C : the complex number field, etc. are followed

in the usual mathematical sense.

CHAPTER I

SASAKIAN MANIFOLDS

§l. Contact Manifolds

Let M2!“-l be a differentiable manifold of dimension

2n4-l (2 3). If M2n+1 carries a global differential

l-form n such that

(1.1) n A (dn)n #’O

everywhere on M, then M2n+1 is said to be a contact

manifold or to have a contact structure. Note that a
contact manifold is orientable.

The contact distribution D is a 2n-dimensional
2n+l

distribution on M defined by
pEM2n+l
where
(1.3) D = {X E T M2n+l:n(x) = O].
P P

The condition (1.1) implies that D is not integrable.

The orientability of M2n+l and D implies that the
line bundle TM2n+l/D admits a cross section 1" on which

M2n+l

n(P) = 1. Thus admits a global non-vanishing

vector field, denoted by 5, such that

(1.4) n(§) = l

and

(1.5) dTl(§oX) = O

for all vector fields x on M2n+l. 5 determines a

l-dimensional integrable distribution complementary to D.
g is called the characteristic vector field of the
contact structure.

From (1.4) and (1.5) it follows that n and dn
are invariant under the action of the l-parameter group

of 5, that is,

(1.6) 8 = O and Q dn = O,

5'” s
where 8 denotes Lie differentiation.

A contact structure is said to be regular if g is

a regular vector field on M2n+l, that is, every point

2n+l

p E M has a cubical coordinate neighborhood 0 such

that the integral curves of g passing through 0 pass

through the neighborhood only once.

It is well-known that a contact structure on M2n+l

defined by a global contact form admits a G-structure
with G = U(n) x 1. In particular, we have the following

theorem. (See Gray [13] and Chern [10])

Theorem 1.1. Let M2n+l be a contact manifold. Then

the structural group of the tangent bundle TM2n+l .9;

M2n+1 can be reduced to U(n) x l.

Examples of Contact Manifolds

JR2n+1

Example 1.1. with the cartesian coordinates

(x1....,xn,yl,...,yn,z). A contact form n and the

characteristic vector fields 5 are given respectively

by
n . .
n = dz — Z) yldx1
i=1
and
-__5_
g — oz '

The contact distribution D is spanned by

i ax1 oz
and , (i = l,...,n)
x ____5__
n+i i '
6y

On the other hand a classical theorem of Darboux says
that a contact manifold always has a local coordinate
system on which n is given as in Example 1.1. (See
Cartan [9]) In particular, we have

Theoremlllg. ‘Lgp w ‘pp_3 l-form on an n-dimensional
differentiable manifold M? and suppose that w A (dw)p i'o
lgpg (dw)p+l = 0 pp_ Mn. Then about every point there
exists a coordinate system (xl,...,xp,yl,....yn-p) ppph,
gag w=dyp+l- ‘57 yidxi.

i=1

Thus: about every point of a contact manifold M2n+l

there exists coordinates (xl,...,xn,y1,...,yn,z) such
that

n . .
i=1

Example llg. 3-dimensional torus T3 considered as a

quotient space of RB. More precisely, consider R3 with

coordinate (xl,x2,x3) and define n = cos x3dx14-sin x3dx2.

Then n A dn = -dx1 A de A dx3. NOW let G be the group

of translation of R3

i = 1,2,3}; then T3 = 1R3/G is a torus which clearly

with generators {x1 4 xii-2w,

carries the contact structure n. The characteristic vector
field 5 of this contact structure is given by

- 3.5.. - 3__?2_
§ - cos x 1 + Sin x 2 .

6x 5x

The integral curve of g through (0,0,g) is given by

x1 = -]-'t, x2 = @t, x3 = 71 Therefore g induces an

2 3'
irrational flow on the 2-dimensional torus x3 = g and
hence the contact structure n on T3 is not regular.

It is known that T3 cannot carry a regular contact struc-

ture. (See e.g. Blair [4]) Also, it is known that every
compact orientable 3-manifold carries a contact structure.

(See Martinet [20])

Example 1.3. M2n+l c: 1R2n+2 with TXMZI‘+1 n {O} = ¢-

The following theorem is well-known. (See Gray [13])

M2n+l 4 1R2n+2

Theorem 1.3. Let i : be a smooth

 

 

hypersurface immersed in R2n+2 and suppose that no
tangent space of M2n+1 contains the origin of 1R2n+2 .

 

 

Then M2n+l has a contact structure.

10

By using the cartesian coordinates (x1,...,x2n+2)
on 1R2n+2, a contact form n on M2n+l is given by
*
n = i a, where
a = xldxz-x2dx;-t..utx2n+ldx2n+2-x2n+2dx2n+l.

As a special case, an odd-dimensional sphere carries

a contact structure. Moreover, since a is invariant

under the reflection through the origin of R2n+2 ,

2n+l(

an
odd-dimensional real projective space ]P R) also

carries a contact structure.

*

l
sphere bundle and the tangent sphere bundle of a Riemannian

Example 1.4. T M, T1M' It is known that the cotangent
manifold are contact manifolds. ‘(See e.g. Reeb [30] and

Sasaki [33])

Example 1.5. Principal Circle bundles over symplectic
manifolds M2n with symplectic 2-form 0 such that
[Q] E H2(M2n, Z) are contact manifolds. (See Boothby-

Wang [8])

§2. Almost Contact Manifolds

In view of Theorem 1.1, we say that a differentiable

2n+l

manifold M has an plmost contact structure if the

structure group of its tangent bundle is reducible to
U(n) x 1.
An almost contact structure can also be seen from a

different view point. A differentiable manifold M2n+1

11

is said to have a (m,§,n)-structure if it admits an
endomorphism. w of the tangent spaces, a vector field g,

and a l-form n satisfying

(1.7) n(§) = 1
and
(1.8) :p2=-I+n®§.

where I denotes the identity transformation.

It is easy to show the following prepositions.

 

 

Proposition 1.4. Suppose M2n+1 has a (m,§,n)-
structure. Then
(1.9) mg = O ‘gpg n 0 w = 0.

Moreover the endomorphism m has rank 2n.

2n+l

Proppsition 1,5. l§_ M, is a manifold with a

2n+l

(m,§,n)-§prpctgrelthen M admits a Riemannian metric

9 such that

(1.10) 9(caX.CpY) = g(X.Y) -n(X)n(Y).

M2n+l with (m,§,n)-structure carrying a metric

satisfying (1.10) is said to have a (m,§,n,g)-structure or
‘ppialmost contact metric structure and the metric g is
called a compatible metric.

By setting Y = g in (1.10), we see that n is the

covariant form of g, i.e.,

(1.11) n(x) = 9(§.X).

12

The next theorem shows that the notions of an almost
contact structure and a (m,§,n)-structure are equivalent.

Theorem 116. If a manifold_ M2n+1

12.5.2. ((1905111)-
ppppcture, then the structural group of its tangent bundlp
is reducible to U(n) x 1. Conversely an almost contact
manifgld carries a (w,§,n)4structure.

In this sense we sometimes refer to an almost contact
structure (w,§,n).

M2n+1

Suppose has an almost contact structure

(m,§,n) with compatible metric g. A 2-form Q defined

on M2n+1 by

§(X0Y) = 9(CPXIY)
is called the fundamental 2-form of the almost contact metric
structure (w.§.n:g). @ is skew-symmetric since
(1.12) 9(ch.Y) = 9(cpzxvcpY) = -9(XocpY)

by (1.8), (1.9) and (1.10). Since m has rank 2n, we
have n A in #0.

The next proposition shows that a contact structure

on M2n+1 gives rise to an almost contact metric structure
on M2n+1.
. . 2n+1 . .
PropOSition 1.7. Let M be a differentiable

 

 

manifold admittingialglobal l-form n and a global 2—form

2n+1

0 such that n A in #’0 everywhere. Then M admits

an almost contact structure. Inlparticular if M2n+1

13

a contact manifold with contact form n, then there
exists an almost contact metric structure (m,§,n,g) pylpp
the same n such that the fundamental Zefppp Q = dn.

An almost contact metric structure with Q = dn is
called a contact metric structure (m,§,n,g), when g
is an associated metric. It is known the associated
metric g of a contact metric structure cannot be flat
for contact manifolds of dimension ‘2 5. (See Blair [5])

Note also that the associated metric is not unique.

Examples of Almost Contact Manifolds

1R2n+1 JR2n+1

Example 1.6. . As we have already seen

carries a contact structure. For convenience, we take as
n . .

our contact form n = %(dz — Z} yldxl) and the character—
i=1

istic vector field = 2‘§%° Then the Riemannian metric

U“

n

g = 33-01% + Z ((dx1)2+ (dyl)2))
i=1
. . 2n+1 .
gives a contact metric structure on I! . The matrix
of components of g is given by
i j _ i
éij+y y 0. y
_ 1
"YJ I O: l 1’

and the (l.l)-tensor field m is given by the matrix

14

O ' y]! 0 0

Note that the characteristic vector field g generates

a l—parameter group of isometries of g, i.e.,

thus i is a killing vector field with respect to g.

M2n+1 c M2n+2

Example 1.7. with almost complex struc-

ture.
Corresponding to Theorem 1.3, we have the following

theorem. (See Tashiro [41])

Theorem 1.8. Let i:M.2n+1 + M2n+2 be a smooth

orientable hypersurface immersed in M2n+2 carrying an

2n+1

 

 

almost complex structure. Then M has an almost con-

tact structure.

2n+1 CJR2n+2

As a special case S carries an almost

contact structure.

2n 2n

Example 1.8. M x II, where M carries an almost
complex structure J. We consider the manifold M2n+l =
M2n x It, though a similar construction can be carried out
for the product M2n x 81. Denote a vector field on M2n+l
by (X,f é%), where X is tangent to M2n' t the coordi-
nate of It and f a C” function on M2n+l. Then taking
n = dt, g = (O,-é% and w(X,f é%~ = (JX,0), we see that

(m,§,n) is an almost contact structure on M2n+l.

15

Example 1.9. A Brieskorn manifold admits a non-regular

almost contact structure. (See Abe [1])

§3. Normal Almost Contact Manifoldp
It is well-known that an almost complex structure need
not come from a complex structure. 86 carries an almost
complex structure that is not complex structure. However,
we have the following theorem of Newlander and Nirenberg.
Theorempllg. .An almost complex structure is a complex
structure if and only if it has no torsion.
Here torsion is the Nijenhuis torsion tensor [J,J]
of the almost complex structure J. In general, the Nijenhuis

torsion tensor is defined for a tensor field h of type

(1.1) as
(1.13) [h,h] (X,Y) = h2[X.Y] + [hX.hY] -h[hX.Y] -h[X.hY] .
If [J,J] = 0, an almost complex structure is said to be

integrable.

Consider the manifold M2n+1 X II where M2n+1

carries an almost contact structure (w,§,n). A vector
2n+1
x

field on M R is denoted by (x,£ 3%) where x
is tangent to M2n+l, t the coordinate of It and f

a C” function on .M2n+1 x 1!. Define an almost complex
structure J on M2n+1 x I! by

(1.14) J(X,f 391;) = (ch-fg,n(x) 25%;.

2

It is easy to check J -I. We say that the almost

16

contact structure (m,§,n) is normal if the almost complex
structure is integrable.

By straight forward computation, we see that the
integrability of the almost complex structure is equiva-

lent to the vanishing of the following four tensors.

N”) (X,Y) = [mac] (X.Y) +dn(X.Y)§.

N‘Z’mm = (wanHY) - (fianHX).
Nmm = (figmx.
N(4)(X) =

(8mm.
5

But we have the following pr0position.

.gpgposition lllQ. For an almost contact structure
(m,§,n) the vanishing of N(l), lmplles the vanishing of
No), N(3) m N(4).

. 5
There is a non—normal almost contact structure on S .

In the case of a contact metric manifold (m,§,n,g)

we'have

Proppsitlon 1.11. Let (@,§,n,g) be a contact metric

plructupel_ Then the tensors N(2) and N(4) vanish.

Nm

Moreover vanishes if and onlyfiif the characteristic

vector field § is killing with respect to g.

In view of Proposition 1.11, we now define a K-

contact manifold. Let M2n+l be a contact metric manifold

2n+1

with structure tensors (m,§,n,g). M is said to be

a K+contact manifolg if the characteristic vector field 5

is a killing vector field with respect to g. It is easily

17

derived that a K-contact manifold satisfies
(1.15) V i = mX,

where v is the Riemannian connection of g.
It is known (Hatakeyama [15]) that a compact regular
contact manifold carries a K-contact structure, but a K-

contact structure need not be regular. (See Example 1.13)

§4. Sasakian Manifolds

If a contact metric structure (m,§,n,g) on M2n+l

is normal, M2n+1

is said to have a normal_contact metric
structure or a Sasakian structure.
It is well-known that we have

Theorem 1.12. An almost contact metric structure

(m,§,n,g) is Sasakian if and onlyiif
(1.16) (VXCp)Y = -g(X,Y)§+n(Y)X

ypppp v denotes the Riemannian connection of g.

A Sasakian structure is in some sense an odd-dimensional
version of a Kahler structure. Thus Sasakian manifolds
provide us with very interesting subject in differential
geometry.

Submanifolds of Sasakian manifolds have been studied by
several people such as Kon, Ludden, Okumura, Tanno and Yano.
(For detailed bibliography, see Yano-Kon [43]). The t0pology
of compact Sasakian manifolds is also an interesting subject.

The first Betti number has been studied by Tachibana [34]

18

and Tanno [36], the p-th Betti number by Blair and
Goldberg [ 6], and the fundamental group by Blair and
Goldberg [6]. Conditions for compact Sasakian manifolds
to be isometric to a sphere are studied by Moskal [24],

Tanno [37], Goldberg [12] and Okumura [27].

Examples of Sasakian Manifolds

JR2n+1

Example 1.10. with the contact metric structure

of Example 1.6 in this chapter is a Sasakian manifold.

Example 1.11. We mentioned that a compact regular

2n+1

contact manifold M carries a K—contact structure

M2n+1

(m,§,n,g). If we consider as a principal circle

bundle by the Boothby-Wang fibration [8] over a symplectic

2n 2n

manifold M, then M is almost Kahler. In this case

it is known (Hatakeyama [15]) that the K-contact structure

is Sasakian if and only if the base manifold M2n is

S2n+1 _’ IPn(C)

Kahler. By considering the Hopf fibration w:
as a special case of the Boothby-Wang fibration, the
usual contact metric structure on the odd-dimensional
sphere is Sasakian.

M2n+1 c M2n+2

Example 1.12. with a Kahler structure.

2n+1 c: JR2n-I-2 a cn+l' the usual

Again by considering S
contact metric structure on the odd-dimensional sphere is
obtained as a Sasakian manifold.

Example 1.13. S3/T, where F is a certain finite
group, has a non-regular Sasakian structure. (See Tanno

[38]) Thus, as we mentioned, a K—contact structure is not

necessarily regular.

Examplegl.14.

19

A Brieskorn manifold is an example

of compact non-regular Sasakian manifolds. (See Abe [2])

Before closing this chapter, we W111 summarize

what we have observed in the following diagram.

 

normal
almost
contact

 

 

 

 

almost
contact

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l
. normal _
compatible > almost -g;;§n—>!Sasakian
metric
contact
metric
X
0,) 3;, a
x \ \
l 00‘ ,«s
l / /'
N(l)= 0 i ,/ /, N(l)_ 0
g 1% /
l as? /
es»
1 I
. 'almost : .._ contact
compatible > contact i ._2;;Qfl_> metric
metric , .
. metric i i e
v 1 o 0’
n A dnnam
with associated
metric

A K—contact structure is between a

structure and a Sasakian structure.

contact metric

S5 carries an almost contact metric structure that

is neither a normal almost contact metric structure nor a

contact metric structure. M

2n

X I1

2n .
where M carries

a complex structure is a normal almost contact manifold

but with dn = O.

For t0pological reasons, the three-dimensional torus

cannot carry a Sasakian structure.

bundles are not, in general, Sasakian.

The tangent sphere

20

The diagram of contact structures corresponds to

the following diagram of complex structures.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

complex g(JX,JY) > Hermitian d0 = 0 > Kahler
4\ ”K /\
[J,J]=0 [J,J]=0 [J,J]=O
almost g(gX,JY) > almost dQ = 0 , almost?
complex = g(X,Y) Hermitian > Kahler

 

 

 

 

 

S6 carries an almost Hermitian structure that is

neither a Hermitian nor an almost Kahler structure. Calabi-

Eckmann manifolds

S2p+1 x 52q+1' p,q 2 1, are Hermitian

but not Kahler. The tangent bundle to a non-flat Riemannian

manifold carries an almost Kahler structure which is not

Kahler.

c“ and 1Pn(¢)

are examples of KEhler manifolds.

CHAPTER II

CURVATURE PROPERTIES OF SASAKIAN MANIFOLDS

§l. Curvatppe of Rlemannian Manifolds

 

Let V be the Riemannian connection on a Riemannian
manifold (Mm,g) which is known to be unique.
Let X,Y and Z be three smooth vector fields in

Mm. Then

VXVYZ - VYVXZ - V [XOY] Z

defines a tensor field of type (1.3). We set

(2.1) VXVYZ-VYVXZ-V[X’Y]Z

= R(X,Y)Z.
R(X,Y) is a tensor field of type (1.1) which is linear in
X and Y. With respect to local components, (2.1) can

be expressed by

. h h _ h i
(2.2) vkvjz -vjvkz — 12k].i z .
We call Rkjih the (Riemann-Christoffel) curvature tensor

of the Riemannian manifold N”.

If we take a l-form m, then we have

(2.3) (vxva) (Z) - (vyvxw) (Z) - ‘V[x,y1‘”) (Z)

= -w(R(X.Y)Z).

21

22

The corresponding local expression is

_ h
(2.4) Vkvjwi'-Vjvkwi - -Rkji wh.

For a general tensor, say T of type (1.2), we have

(2.5) (vaYT)(Z.U)-(vaXT)(Z.U)- T)(Z.U)

(VIXJI

= R(X.Y)T(Z.U)-T(R(X.Y)Z.U)-T(Z;R(X.Y)U):

where U as well as X,Y and Z are smooth vector fields
on ME. The corresponding local expression is

h h _ h t t h
VEViji -vkv£T R T.. -R T

(2.6) ji ‘ zkt 31 ij ti

Formulas from (2.1) to (2.5) are called Ricci identities.
If E1,E2,...,Em are local orthonormal vector fields,

then
.T
(2.7) Ricci(Y,Z) = L; g(R(Ei,Y)Z,Ei)
i=1
defines a global tensor field of type (0,2) with local

components

_ t _ ts
(2.8) Rji - Rtji — g Rtjis
The tensor Rji is called the Ricci tensor.
Moreover, from the Ricci tensor Rji we can define

a global scalar field S by

m
(2.9) S = Z) Ricci(E.,E )
i=1 1 1

or equivalently

(2.10) s = 931 R

The function S is called the scalar curvature. For
surfaces %S is the Gaussian curvature.
From the definition of the curvature tensor, the

following identities are easily obtained.

h h _
(2.11) Rkji A'Rjki — 0,
h h h _
(2.12) Rjki i-Rjik 4-Rikj — 0,
t

or if we put Rkjih = Rkji gth'

(2.14) 0.

Rkjihl'Rjikhl'Rikjh =
Equations (2.12) and (2.14) are Called the first Bianchi
identities.

Applying the Ricci identities to the metric tensor

9, we have

(2.15) Rkjihl'Rkjhi = O-

From (2.13) and (2.15), we have

(2.16) Rkjih = Rihkj'
By using (2.13), (2.15) and (2.16) we see that the Ricci

tensor Rji is a symmetric tensor, that is,

(2.17) Rji = Rij.

From the covariant derivative of the curvature tensor

R, we can prove that

24

(2.18) 0.

VLRkjih + kajLih + ijEkih =
which is called the second BianChi identity.

In the second Bianchi identity (2.18), if we apply

gkh, we get
vR .+VR .h—VR =0
E ji h jLi j Li '
Thus we have
(2 19) v R .h = v R -v R
’ h j£1 j Ei E ji'
Furthermore, applying ng to (2.19), we obtain
h _ L
VhRj - ij VERj ,
that is,
(2.20) v.8 = 2v R.h.
J h)

Let X and Y be two linearly independent vectors
at a point p and p(X,Y) be the plane section spanned by
X and Y. The sectional curvature r(p) for p is
defined by

R(X,YQXJY)

(2.21) r(P) = - 2
g(X,X)g(Y,Y) -g(XlY)

It is easy to see that r(p) is uniquely determined by the
plane section p and is independent of the choice of X
and Y on it.

It is known that the set of values of r(p) for all
plane sections p in TpMm determines the Riemannian
curvature tensor at p of MW.

If r(p) is a constant for all plane sections p in

the tangent space TpMm at p and for all points p E Mm,

then Mm is said to be a ppace of constant curvature.

25

The following theorem is well-known.

Theoremplll. (Schur, 1886) ‘lep Mm be a Riemannian
manifold of dimension m > 2. If the sectional curvature
r(p) depends only on the_point p, ppep .M is a space
of constant curvature.

In this case the curvature tensor is written by

(2°22) Rkjih = r(gkhgji"gjhgki)’
Where r is the constant

S

(2.23) = Efi;fiji3-.

Let E1'°'°'Em be an orthonormal basis at a point

p E MW.

3

(2.24) 1(E1) = .fi) r(P(El:Ei))
1—2
is called the Ricci curvature at p with respect to the

vector El because
m
_ T

r(p(El,Ei)) = Ricci(E
J.

.E )
2 1 1

by (2.21). (2.13) and (2.7).
If the Ricci curvature at p is independent of the

vector Ei' then the Ricci tensor must have the form
(2.25) Rji = lgji .
If this is the case at every point of the manifold Mm,

then the Ricci tensor has the above form anywhere.

If n > 2, by applying 931 and using (2.20), we

see that 1 is a constant. If n = 2, the Ricci tensor

26

always has the above form and l is not necessarily con-
stant. A Riemannian manifold Mm (m‘z 2) Whose Ricci
tensor has the form (2.25) where 1 = a is constant is
said to be an Einstein space.
We call a Riemannian manifold a locally flat space or
locally Euclidean space if the curvature tensor vanishes.
A Riemannian manifold is called a lpgally symmetric

space if its curvature tensor is covariant constant, that

is, VR = 0.

§2. Curvature Properties of K—contact Manifolds

Let M21“.1 be a K—contact manifold with structure

tensors (m,§,n,g). Recall that we have

i i
2.26 v.6 = n.
( ) 35 $3
and
11(3).l = into} = o .
J n J
. . . . 2n+1
USing the Riemannian connection v on M , we have

i,.k i_k1iik

O = 8...” . = v n . V g +n v .5:
54)] S kcpj KP] k Vpk 3b
_ lk 1 . k.n i . i, k _ ,k . i
" a chPj 433' \pk 4-ka \pj — D vk‘pj °
First we have
. . 2n+1 .
Propos1tion 2.2. Let M be a K-contact manifold

 

 

(m,§,n,g). Then the sectional curvature of anyjplane

section containing 5 is equal to l.

27

Proof: Let X be any unit vector field belonging
to the contact distribution D, which is orthogonal to

g for any associated metric. From the Ricci identity

(2.2) we have
h

k-j;i _ . ¢h
Pj(ji X S b Xh ‘ (Vkvjb -Vjvk§ X 5 )(h
h h

k-j

(“wthk§j)Xth = «pijth = 1,

when we used (2.26), (2.27) and (1.8). The left hand

h k-'
) 3

side of the above equation represents the required
sectional curvature. Q.E.D.

As a corollary, we see that on a K-contact manifold
of dimension 2n4-l the Ricci curvature in the direction

g is equal to Zn since

=jei _ 1<=j=i _
(2.28) Rjib b _ Rkji 3 5 — 2n.

The converse is also true giving a characterization of K-
contact manifolds. (See Blair [4])
. . 2n+1 .
Theorem 2.3. A contact metric manifold M is a

K—contact manifold if and only if the Ricci curvature in

the direction of the characteristic vector field g is equal

139 2n.
§3. Basic Curvature Properties of Sasakian Manifolds
Recall that an almost contact metric structure

(m:§.n.g) is Sasakian if and only if

i _ -i i
(2.29) Vkpj - —gkjg +-6k nj.

28

Lemma 2.4. On a Sasakian manifold we have

h :i _ h h
(2.30) iji b — 5k nj-bj nk.
9.1;
(2 31) h = _
° Rkji nh Tlkgji njgki°

Proof: Since a Sasakian manifold is a K—contact
manifold, (2.26) holds. By (2.26) and (2.29), the Ricci

identity (2.2) is

 

h-i -h -h h h
h h -h h h h
= — F ’— — = —

(2.31) is proved by using the Ricci identity (2.4). Q.E.D.
By applying 6hk to (2.30), we have a corollary.

Corollary 2.5.
-i
2.32 R..' = 2n ..
( ) 315 n)
Note that (2.32) also holds on a K—contact manifold.
There is a converse to Lemma 2.4. (See Hatakeyama,

Ogawa and Tanno [14])

Theorem 2.6. Let M2n+1 be a Riemannian manifold

admitting a unit killing vector field g such that
h-i _ h h 2n+1 . . .
Rkji 5 - 6k nj-6j nk, then M is a Sasakian manifold.
By using the Ricci identity (2.6) for mih, we have

Lemma 2.7. On a Sasakian manifold we have

h z 1 h _ . h h
(2°33) Rkji Cpi "Rkji ”1 ‘ ‘fk gji'ij gki

29

Proof: The equation (2.6) for rib is

h h_ h 1 1 h
Vij’wOi 'Vjvk‘Pi "Rkj1 cpi 'Rkji q‘i °

By using (2.26) and (2.29), we get the result. Q.E.D.

Corollary 2.8.

1 kh _
1 1 _
kh _ 1

Proof: By applying 5hk to (2.33), we have (2.34).
. kh . .
Since Rkjihw and mji are skew symmetric in (2.34),
we have (2.35). By using the first Bianchi identity (2.12),

we have (2.36). Q.E.D.

§4. n-parallel Riccl Tensop

First we will prove an identity.

 

 

Proposition 2.9. On a Sasakian manifold M2n+l,
have
-.m.1 . _
(2.37) VjRin"ViRjn — on pi vajE4-4nojinn annjni
n m
“'ZRinmj T)n‘i'ij‘pn nn'

Proof: Consider the both sides of (2.36) as 2-forms
and take the exterior derivative d. First,

cpjidxJ A dxl is nothing but the fundamental 2—form Q

and moreover @ = dn. Thus cpjidxJ A dxl is a closed 2-form.

kh

By applying d on Rkhjiw de A dxl, we have components

 

30

kh kh kh
Vmw‘khjiw H Vj (Rkhimm H vimkhqu) )

kh

kh
‘ C9 (Vijikh + VjRimkh + Viijkh

) + Rj ikhvm‘p

kh kh

+ 4'ijkhviw

Rimkhvjw

k ,h h :k k_h h k
Rjikh('5m b +5m b )+Rimkh(-5j +5j g)

k ch h :k

ch k
(Rjimh + Rimjh + ij ih)° + (Rjikm + Rimkj + ijki)

1f“

:0,

where we have used the second Bianchi identity (2.18),
(2.29), (2.15) and the first Bianchi identity (2.14). Thus
Rkhjimkh dxJ A dxl is also a closed 2-form. Hence

Rjzmiz is a closed 2-form. The exterior derivative of the

2-form Rjflmifi dxj A dxi is expressed by
(2.38) o = v (R. cp.£’)+v.(R. cp ‘)+v.(R 39.“)
m 32 i j 12 m i mfi'j
= Vm(RjLwi£) + Vj (Riflml) " Vi(Rj1cpm£)
= CF’ifi VijN‘me‘VjRu “ViRj1)
+ Rij(—gmi§£+6m}a ni) +Ri (-gjmsz+6j£ nm)
_ RJ£(-gim§£4-éi2 nm)
= $11 VijL+cpm£(VjRj-z - ViRjL)
- 2ngjmni4-ijni '

 

 

31

from (2.35), (2.29) and (2.32). Since (2.32) holds, it

follows that

 

1 _ _ L _ _ 1
(2.39) (vajL)§ - Vm(2nnj) RjLVmg — anmj Rjflmm .
Applying mnm to (2.38) and using (2.39), we have
_ m . L L m _ _
0‘ cpn 3% Vij1+q°m C("n (VjRi1 ViRj ) zmnjni
m
+ ij?n T1i
_ . m L _ _ _ fl 1
_ ¢n mi vnRjL (VjRin viRjn)+2m’°jinn Riz‘oj T1n
-2n +R ~ L -2n» +R ~ m
c[Jijnn jEQi nn anni jmfin ni
=1m~£VR -(VR -vp. )+4n~ -2n~
in 9i m j1 j in i jn wjinn fnjni
. E g m
’ ZRiz‘Oj Tln‘“R;jm"n Tli
which proves the proposition. Q.E.D.
Corollary,2.l9,
(2 40) V R. = w i w £(V.R. -V.R )4-2nw .n -R o L n .
’ m 3n n m 3 1L 1 jz mg n j£ m n

Proof: In the proof of Pr0position 2.9, multiplying

and contract by mnl instead of wnm. Q.E.D.

Corollary lel.

(2.41) (Vg Ricci)(wX,mY) = O.

3'

s 51 to (2.37) and using

Proof: Multiplying mtn m
(2.39), we obtain the result. Q.E.D.
Suppose a Saskian manifold M2n+l has the parallel

Ricci tensor, i.e., V Ricci = 0, then by (2.40) we have

lemm = anmj.

32

Applying mkm, we obtain

Rjk = 2ngjk,

that is M2n+l

is an Einstein manifold. Therefore the
notion of the parallel Ricci tensor of Saskian manifolds

is not essential. But Corollary 2.11 motivates the following
definition. (See Kon [19])

Definition. If the Ricci tensor Rji of the Sasakian

 

 

manifold M2n+1 satisfy
(2.42) (vx Ricci)(oY,nZ) = o

. 2n+1 . .
for any vector fields X,Y and Z pp_ M , then the Ricc1
tensor Rji pp_ M2n+1 is said to be n-parallel.

 

Geometric meaning of the n-parallel Ricci tensor may

be explained from the View point of fibering. Let M2n+l

be a regular Sasakian manifold. If M2n+1

2n+1

/§ denotes the
set of orbits g, then M /§ is a real 2n—dimensional

Kahler manifold. (See Example 1.11, or Ogiue [26]). Thus

there is a fibering' 1T:M?n+l 4 M2n+l/§. Suppose XL, YL
and ZL on M2n+1 denote the horizontal lifts of X,Y
and Z on M2n+l/§ respectively with respect to the

connection n. Then there is a relation

, L
(2.43) (VX Y) = -m2V L YL,
X

where V' denotes the induced Riemannian connection on

M2n+l/§. Hence the Ricci tensor RT. on M2n+1

31 /E is

given by

(2.44) (Ricci'(X,Y))L = Ricci(XL,YL)4-Zg(XL,YL).

 

33

From (2.43) and (2.44), we obtain

(2.45) ((vi Ricci’)(Y,Z))L = (v L Ricci)(YL,ZL).

X

This shows that the Ricci tensor Rji on M2n+l/§ is

parallel if and only if (V L Ricci)(YL,ZL) = 0 Which is

X

equivalent to (VCPU Ricci)(oV,wW) = 0 for any U,V and

W E TPM2n+1 because the horizontal space is spanned by
{wU::U E TpM2n+l}. First (VpU Ricci)(wV,pW) = 0 implies
that

(V 2 Ricci)($V,wW) = -(VU Ricci)($V,oW) = O

:pU
because Corollary 2.10 holds for any Sasakian manifold.

The converse is also true. Thus we have

 

 

Proposition 2.12. Let M2n+1 be a regular Sasakian
. . . 2n+1 v
manifold. Then the Ricc1 tensor Rji [pp M _§_n-

parallel lf and only if the Ricci tepppp Rji ._p M /E
is parallel.

In other words, the notion of n-parallel Ricci tensor
on a Sasakian manifolds corresponds that of parallel Ricci
tensor on a Kahler manifold.

Later we will give a sufficient condition for n-parallel
Ricci tensor.

2n+1

Lemma 2.13. Let M be a Sasakian manifold with

n-parallel Ricci tensor. Then we have

(2.46) VjRim = 2n(:pjinm-:omjni)

2

n 10 I-
' Ri1‘°j Tlm"Rj1‘*°m T‘i'

34
Proof: Since M2n+1

the term mnl sm‘ ijiz of the right hand side of (2.40)

has n-parallel Ricci tensor,

vanishes. Combining (2.37) and (2.40), we obtain (2.46).
Q.E.D.
2n+1 . . .
Corollary 2.14. [ll M is a Sasakian manifold

with n-parallel Ricci tensor, we have

 

 

 

 

(2.47) ijini-viani-vnRji = 0.
. . 2n+1 . .
PropOSition 2.15. Suppose M is a Sasakian
manifold M2n+1 with n-parallel Ricci tensor. Then we
have
(1) the scalar curvature S ‘_f M2n+1
constant,

(2) the square of the length of the Ricci tensor

R.. .pg M2n+1 l§_con§pant. that is,

j].

[Ricci]2 = RjiRJl = constant.

Proof: Multiplying gln to (2.46), we get VjS = O,

which proves (1). For (2), we have

 

 

ji _ ji =
Vk(RjiR ) - 2(VkRji)R O
by using (2.46) and (2.32).
Propositlon g,16. Let M2n+1 be a Sasakian manifold.
2n+1

The Ricci tensop Rji pf_ M .lg n-parallel if and only
lflthe following equation is satisfied

(2.48) [V Ricci]2 = ZIRiccil2-8nS+-16n34-8n2.

 

 

35

Proof: By combining (2.37) and (2.40), we have

=P -" - L
ij. 0 mi Vij£ + 2n(mjinn cnjni) Rifiwj nn

1 , 1 2
Rizmj nn+‘Rj1‘Dn nil

3

2|Ricci{2-8nS+-l6n +8n2

i

. . m
+ terms invo Vin c v. .R .
1 g on 91 V3 m1

But mnm mil VijE vanishes if and only if the Ricci tensor

is n—parallel. Q.E.D.

§5. m—sectional Curvature

In Proposition 2.2 we saw that on a K—contact manifold
the sectional curvature of a plane section containing the
characteristic vector field i is equal to 1. In parti-
cular Sasakian manifolds have this property.

A plane section p in the tangent space TpM2n+1

is called w-section if there exists a vector X 6 TPM2n+1

orthogonal to 5 such that {X,wx] is an orthonormal

basis of the plane section. The sectional curvature

r(p) = r(p(X,mX)) is called a w—sectional curvature.
Similar to the Riemannian case, on a Sasakian manifold

the o-sectional curvatures determine the curvature com-

pletely. (See Moskal [24])

Corresponding to Theorem 2.1 of Schur, we have a

theorem of Ogiue [25].

36

Theorem 2.17. If the m-sectionallcurvature at
any point of a Sasakian manifold of dimension .2 5 ‘lg
independent of the choice of m-section at the point, then

lt is constant on the manifold and the curvature tensor

is given by

 

(2.49) R(X,Y)Z = 25—3-(g(Y,Z)X-g(x,2)Y) + C;1 (n(X)n(Z)Y

- n(Y)n(Z)X + g(X.Z)n(Y)§ - g(Y.Z)n(X)§

+ g(mY,Z)wX - g(pX,Z)mY - 2g(wX,Y)mZ),

where c is the constant m—sectional curvature.
If a Sasakian manifold of dimension 2n4-1 (2_5)
has the curvature tensor of the form (2.49), the Ricci

tensor Rji and the scalar curvature S are given by

 

 

(2.50) Ricci(X,Y) = n(°+3;+°‘l g(X,Y)
and
(2.51) S = %(n(2n4-l)(c4-3)4-n(c-1)).

A Sasakian manifold of constant m—sectional curvature

c will be called a Sasakian spgce form and denoted M21“.l

(C).

Clearly the curvature condition (2.49) is a sufficient
condition for a Sasakian manifold to be a Sasakian space

form.

Examples of Sasakian Space Forms

Example 2.1. 52n+l. The usual contact structure

1

induced on the unit sphere in Cn+ has the metric of

37

constant curvature 1 as an associated metric. We denote
this contact metric structure by (m,§,n,g) and consider

the deformed structure

* *

l
T) =GT): 5 =5
)

m1

I '19 = ‘39:
*

where a is a positive constant. Such a deformation is
called a D-homothetic deformation, since the metrics re-

stricted to the contact distribution D are homothetic.

Tanno showed that S2n+1 with this structure is a Sasakian

space form with constant m—sectional curvature c = £1---3.

a
(See Tanno [38])

Example 2.;. 1R2n+1 . With the structure introduced

in Example 1.6, 1R2n+1'.is a Sasakian space form with
c = -3.

Example 2.3. The product bundle (L,CDn), where CDn
is a simply connected homogeneous complex domain with con-
stant holomorphic sectional curvature < 0 and L is a

real line, gives an example of Sasakian space forms with

c < -3. (See Tanno [38])

§6. n-Einstein Manifolds

The formula (2.50) suggests the following definition.

 

Definition. Let M2n+l be a Sasakian manifold. If
the Ricci tensor Rji of M2n+1 has the form
(2.52) Rji = agjiivbnjni,

38

where a and b (#’0) are constants such that antb = 2n,
then M is called an n-Einstein manifold.
It is easily seen that the scalar curvature of

an n-Einstein manifold is 2n(a+-l).

 

CHAPTER III

CONTACT BOCHNER CURVATURE TENSOR

From this chapter Sasakian manifolds always have

dimension 2,5.

§l. Weyl Confopmal Curvature Tensor
Let Mm be an m-dimensional Riemannian manifold
with metric tensor g and a a positive function on

Mm. Then

* 2
g = a 9

defines a new metric tensor on M Which does not change
the angle between two vectors at a point. Hence it is a
conformal change of the metric. In particular, if the
function a is a constant, the conformal transformation
is said to be homothetic.

Let Mm be of dimension greater than or equal to 4.

The Weyl conformal curvature tensor C is defined by

h_ h h h h h
(3'1) iji ' Rkji + 5k Lji ' 5j Lki + Lk gji ' Lj gki'

where

39

4O

_ _ _;L_. S
(3-2) Lji" 11—2 R'ji‘+ 20n-1J(nw-2) gji

and

(3.3) Lkh = thgth.

The tensor field C is invariant under any conformal
change of the metric.
If a Riemannian metric g is conformally related
to a Riemannian metric g* which is locally flat, then
the Riemannian manifold with the metric g is said to
be conformallylflat.
The following is a well-known theorem of Weyl.
Theorem 3.1. A necessary and sufficient condition

for a Riemannian manifold Mm (m‘z 4) to be conformally

 

flat is that C = 0.
It is easy to see that a space of constant curvature

is conformally flat.

§2. Bochner Curvature Tensor

Let (Mzn,J,g) be a Kahler manifold of complex

dimension n (2_2) with the almost complex structure J
and Kahler metric g.

S. Bochner introduced the so-called Bochner curvature

tensor D on M2n given by

(3.4) D * * = R * *
by Bo 6y Ba

1
-m(R*g *+R *9 *+9*R *+g *R 1:)
y B 60 by Bo y B 6a 6Y Bo

S

(9 g i-g 9 )
2(n+1)(n+ 2) Y*B 601* 5Y* 30*

+

41

with respect to complex local coordinates. (See Bochner
[7])

Then S. Tachibana obtained the real expression of

 

the Bochner curvature tensor. (See Tachibana [35])
h _ h 1 h h h h
(3'5) iji ' Rkji + n+4<5k Lji 5:. Lki+Lk gji'Lj 91(1)
h h h h
+ Jk Mji-Jj Mki+Mk in-Mj Jki
, h h
‘ 2(‘ijMi +Mkj‘pi )'
where
(3 6) L.. = -R . + -——§——— 9
' )1 ji 2(n4—2) ji
and
. _ k
(3.7) Mji — -ijJi .

The Bochner curvature tensor is the formal analogue
of the Weyl conformal curvature tensor. Thus there are
many results corresponding to those in the Riemannian
case. Clearly a homothetic transformation leaves the
Bochner curvature tensor invariant. However, it is still
an open question as to what kind of transformation corres-

ponds to a conformal transformation in the Riemannian case.

§3. Contact Bochner Curvature Tensop

Let M2n+1 be a Sasakian manifold with structure tensor

(w.§.n.g). The contact Bochner cupvature tensor B of

M2n+1

was obtained by Matsumoto and Chfiman in [21] as an
analogue of the Bochner curvature tensor of a Kahler

manifold.

 

M”k M31 ‘thMki‘LMkh mji ‘Mjh wki
‘ 2W’ijih““‘1g"°ih)
-+(wk§ uji-ojh oki-Zokjmih).
where
(3.9) Lji = m (-Rji - (1+ 3)gji+ (L-lmjni).
(3.10) Lji = thgti,
(3.11) L jiLji,
(3.12) M],i —thnit,
and
(3.13) Mji = thgti.

From (3.9) and (3.11) it follows that

 

_ _ s+ 2(3n+ 2)
where S is the scalar curvature of M2n+l.

Applying (2.32) to (3.9), we have
(3.15) L..; = —n. ,
Which, together with (3.12) yields

(3.16) M.

 

 

43

The following identities are easily verified.

(3.17) Bkjihi-Bjkih = o,
(3.18) Bkjihi-Bjikhi-Bikjh = 0,
(3.19) Btjit = 0,

(3°20) Bkjihl'Bkjhi = 0'
(3.21) Bkjih = Bihkj'
(3.22) Bkjih nh O,
(3.23) Bkjth nit = Bkjit ”th'
(3.24) B ..h mkj e 0.

Clearly D-homothetic deformations leave the contact
Bochner curvature tensor B invariant. But again we
do not know what kind of non-trivial transformation leaves

B invariant.

§4. Vanishing Contact Bochner Curvature Tensor

From the point of view of Theorem 3.1, the vanishing
contact Bochner curvature tensor, i.e., B = 0 may give
some geometric meaning.

First we have

Proposition 3,2. Let M2n+1 be a Sasakian manifold.

lfi. M2n+1 2n+1

has constant m-sectional curvaturelpthen M
.ifi n-Einstein and the contact Bochner curvature tensor B

vanishes.

 

 

44

Proof: The first part was already observed in (2.50),
so we will just prove the second part. By using (2.50)

and (2.51) we have

 

 

 

L = _ nc4-Zn4—4 '
and
_ _ c-t3 c-5
Lji ' 8 gji + 8 njnl '
thus
_ c-+3 .

which, substituted in (3.8), gives the result. Q.E.D.
The converse of Pr0position 3.2 is given in the

next proposition.

Proposition 3.3. Let M2n+l be a Sasakian manifold.

If the contact Bochner curvature tensor vanishes and

M2n+1 is an n-Einstein manifold, then M2n+1 has a con-

 

stant w-sectional curvature.

2n+1

Proof: Since M is n-Einstein, the Ricci tensor

is expressed by Rji = agjii-bnjni, where a and b are
constants such that. amtb = 2n. Thus the scalar curvature
S = (2n+-l)a + b is constant. By using (3.8), we can

compute Rkjih' which has the form (2.49) with

= 2na+4a-3n2-5n+2
(n+1)(n+2) '

Q.E.D.
Next, we will weaken our condition on the contact
Bochner curvature tensor and assume that the contact Bochner

curvature tensor is parallel, that is, VB = 0.

45

The following pr0position gives one sufficient
condition for a Sasakian manifold to have a n-parallel

Ricci tensor.

Proposition 3.4. Let M2n+1 be a Sasakian manifold

withpparallel contact Bochner curvature tensor and constant

scalar curvature. Then the Ricci tensor of M2n+l

n-parallel.

Ppppg: By using the curvature properties of a Sasakian
manifold that were given in Chapter II, we obtain the follow-
ing formula straightforwardly.

h _ .
(3.24) Vthji — -2n(VkLij-ijki+-nk(¢ji+-Mji)

+ nj (:oik+ Mik). + 2ni(~1)jk+ Mjk)

1 . h . h . . h
‘ 2(n+2)(‘9ji‘pk “91169;; +2‘9jk‘91 )Vhs)'

By applying pvt muk wt] $81 and making use of (2.40),

we obtain

k
Vsttmu .Dv _ O. Q.E.D.

Another straightforward computation gives a corollary

to Proposition 3.4.

 

 

 

 

Corollary 3.5. Uhder the same assumptions on M2n+l,
the curvature tensor R pf. M2n+1 satisfies
(3.25) (VXR) (”DY/321$VIAOW) .

If the curvature tensor R of a Sasakian manifold

M2n+1 M2n+1

satisfies the condition (3.25), is said to

be n-locally symmetric.

46

§5. Length of Contact Bochner Curvature Tensor

If Mm (m.2 4) is a Riemannian manifold, the square
of the length of the Weyl conformal curvature tensor C
is given by

2 4

m-2

2 2

. . 2
lRiCCi] + (m-1)(m-2) S .

 

(326) [cl =IN2-

If M2n (n.2 2) is a KAhler manifold, the square of

the length of the Bochner curvature tensor D is given
by

2 8

(3.27) (n) =1R: -n+2 2 2

(n+1)(n+2)S

 

[Ricci]2 +

In this section we are going to compute the length

 

of the contact Bochner curvature tensor B of a Sasakian

manifold M2n+1

and study some-applications.

As we see from (3.8), the contact Bochner curvature
tensor consists of 14 terms. Thus when we compute the
square of the length of B, we can expect 142 terms. How-
ever, there are many similar terms.

Let [s,t] 1 g_s, t g_l4 denote the contraction
of the s-th term and the t—th term of the contact Bochner
curvature B. Then, by straightforward computation, we
obtain the following:

{1.1) = IRIZ.
(1.2) (1.3) = (1.4) = (1.5) = Rkjihm

_ l . .2 1 2_3n+2 _2n(2n+l)
— 2(n4—2)(_!RlCCIl 4'4(n4-1) S 2(n+—l) S n+-1 )'

kh k h "
-§ 3 )le

{1:61 -

{1,12}

-(213)

{5,8}

{2.13}

47

[1,7] = {1,8} = (1.9] = %{1,103 = %[1.11}

 

 

kh ji
‘ Rkjih” M
_.___].-__ l- -2 __].-_.___ 2 (2n+3)(n-2)
‘ 2(n+2)(')R1CCll +4(n+l) S + 2(n+1) S
+2n(3n2+5n+1))
n+1 ’
_ _ 1 _ kh_ji _ _ 2

‘12051 = {207] = [209} ‘13141 = {306} = {308]

 

 

 

 

= -{4.5} = {4.7} = (4.9) = {5.6) = (5.8) = -(6:73
= —{6,9} = -{7,8} = ”[809]
= %{s,t} (s = 10,11, 2 g.t $.9)
= é%{u,u} (u = 6.7.8.9)
=-g%{v,v} (v = 10.11)
'h' -'-h ki
— (1.12
31
2
___l____2 ..2£_4__25n+12n+8
— (2(n*_2)) (lRiCCil - . 2 s -+ 2 s
8(n+ 1) 2(n+1)
_8n4+l9n3+l2n2
2(n+1)2
(s = 2,3,4,5)
kh ;k,h ji
(gkh'leTlh)Lji(g ‘b o )L
= 2nlL..[2
3.1.
= 2n(-2—(I-1-]'T-2--)")2(IRicci]2—'—-—'----3n+4 2 82
8(n+1)
5 2 3 2
+ n +l2n+8 S+29n +92n +96n+32)'
2(n+1)2 2(n+1)2

= {3,12} = {4,13} = (5.12]
= -(6.13} = -{7.12) = -(8.13) = -{9.12)

= %{s,t} (s = 10.11, t = 12,13)

48

-.JL _.Ji ._.JL _.Je
_ 2n{6,12} - 2n[7,13] — 2n{8.12) - 2n{9.13}

= {u,l4} (2 g u g 9)
= 515(v,14} (v = 10.11)

_ _ _ fljhlki
' (gkh T‘k'lhn‘ji‘o @
= L.l+l
l
_ _ ___;L___ s.____11___
4(n+—l) 2(n4-l) '

{2.4} = (3.5} = {6.8} = {7.9}

 

 

= %{10,11]
= ((2,13}2)
_ i 2
— (Li +1)
2
= 1 2 82+- n 2 S4— n 2 ’
16(n4—1) 4(n4-1) 4(n4-1)
(12.12) = (13.13) = %(14.14)
_ kh ji
wkhmjim w
= 4n2,
-[12,13} = %(12,143 = %{13,14}
_ p . Q0jh ki
‘ thfji- w
- i._
— 51 1
= .211,
[2,6] = {2,8} = [2,12] = [3,7] = [3,9] = [3,13]
= [4.6] = [4,8] = {4,12} = {5,7} = {5,9} = (5.13}
= 0.

Since [s,t] is symmetric, the above pairs are

enough to compute (BIZ.

 

49

 

 

 

 

(3.28) {1312: Z [s,t]= (Rlz—nfz (Riccil2
lgs,tgl4
2 .
2 2 4(3n +3n-A
+(n+1)(n+2)S + (n+1)(n+2)S
_ 4n(6n3+9n2-n-2)
(n+1)(n+2) '

The same result is obtained independently by D.
Janssens [16].

As a preparation we prove the following lemma.

 

 

 

 

Lemma 3.6. .peg M2n+l be a Sasakian manifold. Then
we have
(3.29) (Ricci)2 2.L§JE%ElE-+ 4n2.
The eqtlality holds if and only if 142“” liar; n—Einstein
manifold.

‘gpppf: At each point p of M2n+l, choose an

orthonormal basis including the characteristic vector field

6 of TpM2n+1 so that the matrix representing the Ricci
tensor Rji is diagonalized. Then the scalar curvature is
expressed by
2n
S = 123 Rii4-2n

since Ricci(§,§) = 2n.
By using Schwartz inequality, we get

2 2n
(S-2n) =(Z Rii)

i=1

2n
2 g 2n 2 Riiz = 2n([Riccil2—4n2).
i=1

giving the inequality. The equality holds if and only if

the Ricci operator restricted to the contact distribution

D is gjigfl I, where I denotes the identity. This

 

50

means R. - ELLEE-

3i — 2n gji for l g.i, j g_2n. Since

Ricci(§,§) 2n, we have

_ S-—2n ._S-2n

Now we establish an inequality involving the curvature
tensor R and the scalar curvature S.
2n+1

Proposition 3.7. Let M be a Sasakian manifold.

Then the following inequality holds.

 

2 2 2 __ 4§3n+12 4n(3n+l)(2n+LL
(330) [RI 2;fi;ITTS n+1 5*" n+1 '
The equality holds if and only if M2n+1 has a constant

 

 

w-sectional curvature.

 

 

Proof: First we rewrite [BIZ so that we can use

 

(3.29).
2
2_ 8 4 . .2 m-zm 2
(3.31) [B] - — n4-2 ([RlCCil - 2n - 4n )
2 2 2 4§3n+-12
+ IR‘ - n(n+—l) S + n4-l S
_ 4n(3n+ 1) (2n+ l)

 

ni-l

Now by virtue of [BIZ 2 O and (3.29), we have the result.

M2n+1

When the equality holds, is an n-Einstein

Sasakian manifold with vanishing contact Bochner curvature
tensor. Thus by Proposition 3.3 M2n+l has a constant

m-sectional curvature. The converse is also true by

Proposition 3.2. Q.E.D.

 

CHAPTER IV

SPECTRUM OF RIEMANNIAN MANIFOLDS

§l. Laplacian
Let (Mn,g) be a compact connected orientable [3
Riemannian manifold without boundary with a Riemannian

metric g and let Ap(M) denote the set of p-forms on Mn.

 

The Laplacian A, acting on the real valued C°°— B

functions on M (= C”(M) = AOLM)), is defined by -
(4.1) Af = 6df.

where 6 is an Operator on A1(M) with values in AO(M)

such that 60 = -div 0*, a E A1(M), a# given by

g(d#,X) = d(X), and d is the exterior derivative.

There are other definitions of the Laplacian A.

such as
(4.2) (Af)(p) = -trace(vdf) = —trace(Hessian f)
n
= -.Z; Hessian f (Ei,Ei),
i=1

[Ei] being an orthonormal basis of TpMp, or

n d2(f o vi)
(4.3) (Af)(p) = - Z 2 (0).
i=1 ds

 

{vi} being an orthonormal set of geodesics parameterized

by arc length 5 passing through the point p at s = O.

51

 

52

For computation the following formula is convenient,

(4.4) (Af)(p) = - l E 5(qqu(of/ole

9 i,j=l axl

where g = det(gij) and {xi} is a local coordinate system
on Mn around p.

If f depends only on the distance d(p,p0) between
p and a fixed point p0, i.e., there exists a function

F : JR 4 JR with f(p) = F(d(p,po)), then inside the ball

of center p0 and radius 6 we have

 

(4.5) Af=-—--—(—+

where ' denotes the first derivative with respect to e
and e is a Ca—function representing the ratio between
the measure of the manifold induced by the Riemannian
structure and the Lebesgue measure of TpMp transferred
by the exponential mapping expp.
§2. Spectrum of Riemannian Manifolds
Let (Mp,g) be an n-dimensional Riemannian manifold
and C°(M) be the set of real valued Cm-function on Mn.
The spectrum of (Mn,g), denoted by Spec(Mn,g), is
the set of eigenvalues A of A. i.e., the A's 6 II
such that there exists f e c”(M), f g'o with Af = 1f.
A function f e c°(M) such that Af = xf with
x E Spec(Mp,g) is called a Eggper function associated to
l. The subspace of C”(M) formed by proper functions

associated to X is called proper subspace associated toqk

 

 

53

and is denoted by @X(Mn,g). 9(Mn.g) = Z} n @x(Mp,g)
x E Spec(M ,g)

is called the proper subspace of C°(M), where the sum is

direct sum and 9X(Mp,g)'s are orthogonal to each other in

the sense of the inner product of the pre-Hilbert structure

of c°(M), i.e.,

<f,g> = F f -g v ,
JM 9

vg being the canonical measure on Mp induced by g. i

We write

Spec(M“.g) = {0 = x0 < M 3 x2 gm 3.

 

each A being written a number of times equal to its H
.multiplicity. mul(l) = dim.€i(Mn,g).
The spectrum does not start before 0 because A is

. ' . 2 I
t p o o f ° Af = df d t 1].
p051 ive i e , IM vg IM I l Vg' an i rea y

starts at O with multiplicity one since Af = 0 implies
df = O, i.e., f should be constant and conversely.
(Note <f,Af> = <f,odf> = <df,df>).
Note that Spec(Mp,g) depends only on the Riemannian
structure of (Mn,g).
The following theorem is essential.
Theorem 4.1.
(l) The spectrum sequence lo.ll,l2,... is discrete
and tends to +o.
(2) For every x e spec(M9,g), dim 9K(Mn,g) is
finite.
(3) €(Mp,g) is dense in CQ(M) in the sense of

the topology induced by the inner product < , >.

 

54

Examples
Example 4.1. Ml, i.e., a closed curve of length L.

The eigenvalue of A are

each of multiplicity is 2 except A0 = O.

NOte that by obtaining the spectrum, or even Al, we
know the length of the curve, that is, we know the Riemannian
structure of M1.

Example 4:2. (Sn,g). An n-dimensional sphere with

the usual Riemannian structure gO induced from Iqu'.

k
9xk(5n.go) = 37k.

where Mk C CQ(IJH11 is the space of harmonic homogeneous
polynomials of degree k on Igfiq' and Qk is the
restriction of Wk to Sn, i.e., gk C C°(Sn). We have
an orthogonal decomposition of 9k;

92k = 212k e 1:2?!qu o...<+> 135/0.

92k+1 = ”n+1 e r22IZk_l e...o erivl.

Then we see that the multiplicity of lk is given by

. n
mulek) = (n+k-2)(n]:'k-3)...(n+l) (n+2k-l).

ExampLe 4.3. (1Pn (JR) ,go). An n-dimensional real
projective space with the Riemannian structure induced from

that of sn by the Hopf fibration; (Sn,go) a (113m (1R),go).

55

A
Let “2k be the space of functions on (IPn(1R):gO)
induced by the functions §2k on (Sn,go) through the

Hopf fibration. Then we have

Ak = 2k(n4—2k-—l), k‘z O,

. n A

and

mu1(xk)= fl+2k1§115rm+1m (n+4k-l).

Example 4.4. (H§1(C),go). A complex n-dimensional

complex projective space with the Riemannian structure in-
2n+l 2n+1

 

duced from that of S by the HOpf fibration; (8 ,go)
n
lk=4k(n+k), kZO,

n .A

and
mu1(lk) = n(n4-2k)(n(n4-l)'ii(n+-k-1))2:
where HR k is the space of harmonic bihomogeneous polynomials
I
. - n+1 A .
of degree (k,k) in z and z on C and wk k is
induced by the restriction of Wk k on S2n+1 through
I

the H0pf fibration ¢n+l :3 521141 4 En (C).

Example 4.5. (Tn,g). An n-dimensional flat torus
defined by Tn = Rn /A and g = gO/A, where A is any
lattice of if} and 90 is the usual Riemannian structure

*
on Rn. Let A = {yEJRn: (xly) 6%,XEA) where

56

*
( I ) is the inner product on fifl. A is called

the dual lattice of A.
*
Spec(Tn,g) = [4wlyl2 :y E A ],

9A(Tnog) = {fx : fx(y) = eZTTi(le)' k = 47TIY]2}I

mul(l) = 2 if A > O.

For n = 2, the spectrum of a flat torus determines its

 

 

 

Riemannian structure, that is, F:
Proposition 4.2. Suppose A and A' be two lattices
_f_ IRZ. ;£_f_ Spec(]R2/A,gO/A) =Spec(]R2/A',gO/A'), then
2 2 . . _
(R /AogO/A) = (R /A 090/1) ). W
However, Milnor's counter example [23] shows that

16

there are two non-isometric lattices in E! which give

the same spectrums to the two tori. Also there is a counter
example for T12. (See M. Kneser [18]) From n = 3 to 11

the question is open.

§3. Asymptotic Expansion

The real heat equation on (Mn,g) works with functions
G :M x JR+ 3 (p,t) 4 G(p,t) 6 1R satisfying the two
conditions

99.-

at-O' G(.'O) =f'

AG +

where A is the Laplacian acting on M? and f :Mn 4 It

is a given initial condition.

57

Definition. The fundamental solution of the real

heat equation, denoted by FSRHE, .pp (Mn,g) is a function

 

F:MxMx]R:-o]R
such that
(1) F ip. CO ip the fgpst variable, C2 in the
second and C1 in the third, and then

.222.
A2F + at ’ 0'

where A2 is the Laplacian on the second variable,

(2) for every, p E M, 1im F(p,',t) = 6 , where
t40+ p

op is the Dirac distribution at p.

 

Theorem 4.3. A FSRHE exists and is unique.

Corollary 4.4. There is an asymptotic expansion

(4.6) F(p,p,t) 4 (4Ft)-n/2 (u (p,p)4—tul(p,p)+u..
t40+ 0 k

+t uk(plp)+ooo) I

where ui E C"(M x M).

Since G(Mn,g) is dense in C"(M), we can choose an
orthonormal basis [mi] such that the mi's are prOper
functions. As a Fourier series every f E C°(M) may be
written in the form f = Zj<wi,f>mi in the L2-sense.

Then we have 1
Proposition 4.4. 3;. [mi] is an orthonormal set of

proper functions, then for every p,q and t, the series

Ele wi(p)wi(q)

converges and
,\ -Ait
F(p.q.t) = 4.e wi(p)wi(q).
i

 

58

Corollary 4.5. For every t > O the series
-)..t
ZLe 1 convepges and
i

(4.7) :e i =J" F(p.p.t)vg.

For every i, if we set
ai = [M ui(p,p)vg,
(4.6) and (4.7) lead to

Theorem 4.6. (Asymptotic expansionl, For every

Riemannian manifold, there exist ai's (i = 0,1,...)

with

-x.t _ h .
(4.8) Zle l = (4vt) n/2 Z, ait14-0(t
i i=0

k+l-n/2)

for every k.

§4. Geometric Appllgation of Asymptotic Expansion

Immediately, we see that if (M,g) and (M’,g’) are
two Riemannian manifolds with Spec(M,g) = Spec(M’,g’),
then dim.M.= dim M’.

Theoretically we can compute the ai's but so far

only ao,al,a2 and a3 have been computed.
(4.9) a0 = IM v9 = volume of (Mp,g),
(4 10) - -1- “ s
_ 1 2 . . 2 2

(4.11) a2 - EEE'IM(ZIR] -2[Rlcc1] 4—58 )vg,
and

l .
(4.12) a3 = ET-f fvg, (Sakai [31])

59

where
142 2 26 . . 2 1 2 s 2
(4.13) f = - gig-(vs: --—-lv R1CC1] -§ IVR] 4-§ s
2 2 4 ' ° h
- § isicci]2 +- SIR] -— th 12.3.1 i
_ _2_O ki jh __8__ ts jih
63 R R Rkjiho R th stih

.31 mt
‘ 21 Rih Rmzk JRkj1h

(Note that Sakai's curvature tensor has the opposite sign.)

By comparing a we have

0'

Theorem 4.7. .££ Spec(M,g) = Spec(M’,g’), ppgp
vol(M,g) = vol(M’,g').

For the 2-dimensional case, if we use the Gauss-Bonnet
formula va(M)= I Svg , where X(M) is the Euler
characteristic of 3, we have“

Theorem 4.9. .3; Spec(M,g) = Spec(M’,g’) Egg
dim M.= dim M3 = 2, pp§p_ x(M) = X(M’).

By using a2 and some geometric inequalities, we have

Theorem 4.10. lg, (M,g) is a surface with constant
scalar curvature S and if (M',g’) is a Riemannian mani-
gpld such that Spec(M7,g’) = Spec(M,g), ppgp (M’,g')
is a surface with constant scalar curvature S.

And

Theorem 4.11. .Li (M,g) l§_3’3-dimensional Riemannian
manifold with constant sectional curvature r and if
(M’,g’) is a Riemannian manifold such that Spec(M',g’) =

Spec(M,g), then (M’,g') is a 3-dimensional Riemannian

manifold with constant sectional curvature r.

 

 

60

For 4-dimensional Riemannian manifolds, we have
X(M4) =-—J=§ I (]R]2-4]Ricci]2+-Sz)v .
32w M 9

Hence we get

Theorem 4llg. .lg (M,g) l§_g,4-dimensional;Riemannian
manifoldlwith constant sectional curvature r and if
(M’,g') is a Riemannian manifgld such that Spec(M’,g’) =
SpeC(M.g) Eng X(M') =X(M). 1:1ng (M',g’) i_s_et_4—
dimensional Riemannian manifold with constant sectional
curvature r.

Sakai [31] proved the following two results.

Theorem 42l;. If Riemannian manifolg§_ (M,g) ‘gpg
(M',g’) are Einstein spaces and aB = a’B (B = 0,1,2),
ppgp (M,g) has constant section§l_curvature r if and
only if (M’,g’) has constant sectional:curvature r.

Theopem 4.14. ‘lfi (M,g) QQQ, (M',g') are 6-

dimensional Einstein spaces satisfying a = a . (B = O,l,2,3)

B B
33g. X(M) = X(M'), pp2p_ (M,g) is locally symmetric if
and only i; (M',g’) is lpcally symmetric.

The following result was proved by Mckean-Singer [22]
for n g.3 and Patodi [29] for n = 4,5.

Theglem 4.15. .Lgp (MF,g) ‘p§_§p. n (g_5)-dimensional
Riemannian manifold with a1 = a2 = O. Tppp (M,g) .ii
locally flat.

Tanno [39] obtained the combination of Theorems 12,

13 and 14.

61

Theopem 4.16. Let (Mpag) Egg, (M'n .9’) 22.

compact orientable Riemannian manifolds. Assume Spec(MR,g)
= Spec(M'n ,g'). Then n = n' and
(l) for 2 g.n‘g_5, (M,g) is of constant

sectional curvature r, if and only lg

(M',g') is of constant sectional curvature

(2) For n = 6, (M,g) is conformallylllat and
helgcalar curvatgpg_ S lg constant,ll§
and only ig (M’,g') lg conformally flat
gpglthelggalar curvatppg 8’ lg constant,
lijhich case S = S'. Algg, (M,g) lg_p£
constant gectional cprvature r > 0, ‘ll
and only ll, (M',g’) lg of constant section—
al curvature r' = r > 0.
Corollary 4.17. lgp, (Mn,g) be a compact simply
connected orientable manifplg, 2 g,n g_6. ll. Spec(Mn,g)
= Spec(sn(r),g0), ppgp (M,g) is isometric to (Sn(r),go).
Tanno [39] also proved this type of theorem in the
Kahler case.
Theorem 4.18. .Lg; (M,g,J) Egg (M’,g',J’) ‘pg
compact Kahler manifolds with dime M = 2,3,4,5,6.
Assume Spec(M,g,J) = Spec(M',g',J').
(1) 22$. dimC M.= 2,3,4,5, (M,g,J) lg of constant
holomorphic sectional curvature h, if and
only if (M',g',J’) is of constant hplomorphic

sectional curvature h’ = h.

62

(2) For dimmM.= 6, (M,g,J) is of constant holomor-

 

phic sectional curvature h # 0, if and only

 

ll, (M’,g',J') is of constant holomorphic
sectional curvature h’ = h.
Corollary 4,19. Let (M,g,J) pe a complex n-

dimensional (2 g n g 6) Kahler manifold. Let (1Pn(C),

 

gO'JO) be a complex n-dimensional projective space with
the Fubini-Study metric of constant holomorphic sectional
curvature h. ‘ll Spec(M,g,J) = Spec(n¥1(C),gO,JO), then

(M,g,J) lg'holomorphically isometric to ‘RR1(C)'90'J0)°

CHAPTER V
SPECTRUM OF SASAKIAN MANIFOLDS
Let M2n+l be a 2n4-l (2 5)-dimensional compact
Sasakian manifold with structure tensors ($.i.fi.g).
First we prove
Proposition 5.1. Lgp, (M,g) gpg. (M’,g’) 'pg

Sasakian manifolds of dimension ‘2 5. Assume that

 

Spec(M,g) = Spec(M’,g’), M is op constant m-sectional

curvature c Egg. M’ is an n-Einstein manifold. Then

M’ is of constant m-sectional curvature c’ = c.
Remark. The assumption on the spectrum can be

weakened as a0 = a0, a1 = a1, a2 = a2 if dim M.= dim M .

Proof: Since a0 = a0 and a1 = a1, we have
5.1 V = v'.
( ) (M g (M, g .
and
(5.2) f Sv = J)" s'v'.
M g M’ 9
As for a2, we have

 

 

2 . . 2 2
(5.3) a2 = fM (2]R] -2[Rlcc1] 4-58 )v9
2 2 2 4S3 4'1]
= IM [2(IR] - n(n4-l) S + nfi-l S
2
4 4'1 2 4-1 . . 2 S-2 2
- nLBnn4_i( n )) - 2([RlCCil - ( Zuni—w-4n )

63

64

 

, 4 __l 2 __80n+l)
+ (5 + n(n4-1) n)s + ( n4—1 + 4)S
8an+lfl2n+lL_

+ (

2
n4-1 4n-8n )]vg.

By considering the assumptions, (5.1), (5.2) and

 

 

Lemma 3.6 and PrOposition 3.7, we can express a2 = a5
by
4 _ l. 5‘ a2
_ I 2 _ 2 ’2 4Qn+l) r
' .JFM, 2(IR] n(n+ 1) S + n+1 S
_ 4n(3n4—l)(2n4-1))V.
n4-l '
4 _ 1 ,2 .
+ (5 + “n(n+1) me’ 5 v9. .
Because (3 30) holds and 5 + -——é—-—-- ; > O (5 4) is
' n(nd-l) n ' '
now
2 .2 .
5.5 S v S r .

On the other hand, since S is constant
(= %(n(2n+-l)(c4-3)+-n(c-l))), by using Schwartz's

inequality we have

- 2 I t 2 n 2
5.6 R v’. S' v . ( S v'.) = (‘ Sv )
( ) Jm’ 9 fM' 9 2 JRM' 9 JM 9
2 2 2
= S ( v ) = S v v’. .
IM 9 (M (Jim. 9

Taking account of volume M' > O and (5.5), we see that
the equality holds in (5.6), which implies that M7 has
a constant m—sectional curvature by Proposition 3.7.

Moreover S = S' = constant, and so c = c'. Q.E.D.

65

Qprollarv 5‘3. There lg no n-filnstein Sasakian
manifold of dimension 2_5 with vanishing. a2.
Proof: The sum of the last three terms of (5.3)
is always positive. The first term is positive and the

second term is 0 because of n-Einsteinness. So
a2 > O. Q.E.D.

Pro sition . .Lgp (M,g) gpg_ (M’,g’) be compact
Sasakian manifolds with dim M = 5,7,9,ll,13. Assume
Spec(M,g) = Spec(M’,g').

(1) £9; dim M.= 5,7,9,ll, M is of constant

m—gectional curvgture c, if and only if
M’ ls oflconstant m-sectionalpcurvature
C' (= c).

(2) Egg, dim M.= 13, the contact Bochner curvature
tensor B .9; .M vanishes and the scalgp
curygture S ‘9; M is constant, if and only
lf the contggp Bochnep curvature tensor B’

.9; .M' vanighes and the scalar curvature S'
_l_ M’ lg:copgp§nt.

Proof: By using the contact Bochner curvature tensor

B, we can express a

2, in general, by

(s — 2n)2 _4n2)

 

 

a --——1 [213124. “6'19 (lRiccilz-

 

 

 

 

2 _ 360 (n4-1)(n4-2) 2n
4 6-n 2
+ (5 - (n4—l)(n+—2) + n(n4-1)(n4-2))S
+ (_ 8(3n24-3n-2) 4(6-n) )S

(n+1)(n+2) - (n+1)(n+ 2)

66

8n(6n3 + 9n2 - n - 2)

(n+1)(n+ 2)

2
8n (6-n)
(n+1)(n+2)” Vg°

4n(6-n)

+ ( (n4-l)(n+-2)

+

First Spec(M,g) Spec(M',g') implies dim M.= dim M3.

(1) Suppose M. has a constant m—sectional curvature

 

 

 

c and dim M.= 5,7,9,ll. Then a0— = a0, a1 = a1 and
a2 = a5 gives
4 6-
ULB) 6‘-(n+1Hn+2)+ Ifin+lHn+2fl IM Szvg
=r' ,.2 2(6-n) . ..2
JM’ [Z‘B l + (n4-1)(n4-2) ([Ricc1 I

 

2 .
2n - 4n )]v9.

4 )
(n4-1)(n4-2) +n(n+1)(n+2) IM

 

+ (5 -

By using the same technique used in the proof of

Proposition 5.1, we have (3'! = O and [Ricci’l2 =

. 2
(S -2nL+.4n2’
2n

m-sectional curvature c' = c by Lemma 3.6 and Pr0position

which implies that M' ‘has a constant

3.3. Similarly the converse is also true.

(2) Let dim M.= 13, B = O and S' be a constant.
Then we have (5.8) with n = 6. Hence 8' = O and
S' = S = constant. Q.E.D.

Next we will extend the part (1) of PrOposition 5.3
to dim M = 13 by using the expression of a3 and the
part (2) of Proposition 5.3.

2n+1

Lemma 5.4. Let M be a Sasakian manifold with

constant scalar curvature S and parallelicontact Bochner

tensor. Then we have

3 2

(5.9) [v Ricci]2 = 2]Ricci[2-8nS4-l6n 4—8n

67

Proof: By Proposition 3.4, the Ricci tensor R.

3i
M2n+1

of is n-parallel. (5.9) follows immediately

from Proposition 2.16. Q.E.D.

Lemma 5.5. Let M2n+1 be a Sasakian manifoldpwith

constant scalar curvature S and parallel contact Bochner
curvature tensor. Then we have

(5.10) ]VR]2 = 4]R]2-l6S+32n2+16n.

If the contact Bochner curvature tensor vanishes, we have

32

n4-2 [Ricci]2 - 8 2

(n4—1)(n4—2) S
4(3n2+3n-2)
(n+1)(n+2)

4n(6n3 + 9n2 - n -

(n+1)(n+ 2)

(5.11) ]VR]2 =

( + 16)s

21 + 32n2 + 16n .

M2n+1

Proof: Under our assumptions, is locally
n-symmetric by Corollary 3.5.
From (2.30), we get
i - p i n r
(5.12) (szkjih)R - ‘Rkjith 4'9kh4uj ’ 9jh92k '
and hence
_k-i

Thus we get (5.10). (5.11) follows by substituting (3.28)
into (5.10). Q.E.D.
The following lemma is immediate from (3.28).
2n+1

Lemma 5.16. Let M be a Sasakian manifold with

vanishlng contact Bochner curvature tensor. Then we have

68

2 S3
(n+1)(n+2)
2 3 2
_ 4(3n +3n-2) S2 + 4n(6n +9n -n-2)
(n+1)(n+2) (n+1)(n+2)

Let (Ricci3) denote RhJ le Rih. We have the

8
n4-2

 

(5.14) SIR]2 = SlRicciIZ _

S.

following lemma.

Lemma 5.7. Let M2n+l be a Sasakian manifold with

 

n-parallel Ricci tensor. [Then we have

Rkh 3 2

(5.15) (Ricci3) = le'Rkjih +]Ricci]2-4nS+8n +4n .

Pppgf: The Ricci identity for the Ricci tensor Rji
is given by

(5.16) VijRih"VijRih = RkjlhRi£"Rkji£ Rih '

Applying gki and making use of (2.46), we get this result.

Q.E.D.

In addition to the assumptions of Lemma 5.7, if the

 

 

 

 

 

contact Bochner curvature tensor B of M2n+1 vanishes,
then by using (3.8), Rji RkjihRRh is written by
(5.17) Rji Rkjithh = n42-2 (Ricci3) + 2(n2+nf)]n+ 2) SIRiccilz
{(21:31:32) IRiCCllz
l 3 5n4—4 S2

S +

" 4(n+1)(n+ 2) 2(n+1)(n+2)

3n2 2n3(2n4-1)

+ (n+1)(n+2) S ' (n+1)(n+ 2) °

Substituting (5.17) into (5.15), we have the following

lemma because of Proposition 3.4.

69
2n+1 . . .
Lemma 5.8. Let M be a Sasakian manifold Wlth

vanishing contact Bochner curvature tensor and constant

scalar curvature. Then we have
,. .3 _2n+l . .2 l 3
(5.18) n(RiCCi ) — __2(n+ l) SlRiCCi] - _——4(n+ 1) S

_ n(4n4—5) [Riccilz + lpgtg___52

 

 

n+1 2(n+1)
_ n(4n24-9n4-8) S
n4-l
2n2(4n34-12n24-13n4-4)
+ J: .
n4-1

 

Now we are going to express the last three terms of

(4.13) in terms of (RicciB), {Ricci|2, S and n. The

following lemma is easy to prove because of (5.15) and

Rkjih = "Rjkih°

Lemma 5.9. Let M2n+1 be a Sasakian manifold Wlth

n—parallgl Ricci tensor. Then we have

kijh __..3 . .2 3 2
(5.19) R R Rkjih - (RlCCl )4—(R1cc1l -4nS+-8n 4—4n .

Lemma 5.10. Let M2n+1 pepa Sasakian manifold with

vanishing contact Bochner curvature tensor and constant

scalar curvature. Then we have

tsjih _8 ..3 2 ..2
(5.20) R Rt stih — n4-2 (Ricc1 ) - (n+-l)(n4-2) SIRiCCi]

 

2(5n4-6)(n-1) . . 2
- (n+-l)(n4-2) lRiCCil
4 S2
(n+ l)(n+ 2)

+ 2(4n3+5n2+7n+ 21 s
(n+1)(n+2)

_ 4nl8n24-5n4-2)
n+-2

70

Proof: Since the contact Bochner curvature tensor

vanishes, we rewrite R by using (3.8). We then

sjih
write out the left side by using most of the curvature
properties of Sasakian manifolds and Lemma 5.8. Q.E.D.

Lemma 5.ll. Let M2n+1

be a Sasakian manifold with
vanishing contact Bochner curvature tensor and constant
scalar curvature. Then we have

mt kj ih
(5.21) Rih Rm]! Rkj

— giflitél-(Ricci3) + 2(nj;lQl SlRicci]2

 

 

 

(n+2)2 (n+1)(n+2)2

_ 2 __(rn3 -13n2 -4n+ 28) [Riccilz
(n+ l)2(n+ 2) (n+ l)(n+ 2)
4(2n3+12n2 +9n- 4) S2

 

(n+ 1)2 (n+ 2)2

_ 41g ln 5+es9n4+135n3+132n2+230n+321S

(n+ l)2 (n+ 2)2

_ 8n(8n4 - 8n3 - 59h2 -247n- 2).

(n+ l)(n+ 2)2
Proof: By using (3.8), we rewrite Rkjlh. Then again
we use most of the curvature prOperties of Sasakian manifolds,

(5.28) and Lemma 5.9. Q.E.D.

We now apply Lemmas 5.4, 5.5, 5.6, 5.9, 5.10 and 5.11

to the formula for a (4.12) and (4.13). The coefficient

3I
of (RicciB) is

64(n4-6l, 64 20

(5.22) - ‘—
21(n4-2)2 63(n+2)+ 63

\llvb

71

By using Lemma 5.8, we can now express a3 in terms of

the Ricci curvature tensor, the scalar curvature S and

the dimension of the manifold.

___1_ 32(3n2+20n+lol 8(2n+1)
(5'23) 9 ’ I H 2‘63(n+l)
M 63n(n+—l)(n+-2)

 

 

 

2(6-n) . . 2
+ 3(n4-1)) SlRicc1]
+ (16(17n3-99n2-324n-164)+ 16(in+5)
. 2 63(n+l)
63(n4-l)(n+—2)

32 _8 . ..2
- 9(n4-2) 7) lRlCCl‘

+ cl(n)S3 + c2(n)82+ c3(n)S + c4(n)]vg ,

where the ck(n)'s k = l,2,3,4 are algebraic expressions
in n.

For a lB-dimensional Sasakian manifold with vanishing
contact Bochner curvature tensor and constant scalar

curvature, (5.23) is now

. ___L l ..2_736 ...2
(5.24) a3 - 6! IM (147 SIRiCCi] 441 [Ricc1I

3 2
+ c1(6)S 4-c2(6)S 4—c3(6)Si-c4(6))vg.
Proppsition 5.lg. Let (M,g) and (M',g’) .pg
compact Sasakian manifolds. Assume that dim M = 13 and

Spec(M,g) = Spec(M',g’). Then M is of constant @-

 

 

sectional curvature c #“%%% , if and onlyyif M’ is of
constant m-sectional curvature c' = c.

 

 

Proof: dim M' = 13 since M and M7 have the same
spectrum. Suppose that M is of constant w-sectional
curvature c. Then the contact Bochner curvature tensor

B of M vanishes and the scalar curvature S of M is

72

constant. By Pr0position 5.3 (2), the contact Bochner
curvature tensor 8’ of M' vanishes and the scalar

curvature S' of M' is constant. Thus the condition

a1 = a1 is now

(5.25) S V = S' v'. .
IM 9 (M, g

The condition a0 = a6 implies S = S’.

(5.24) is also expressed by

 

2
(5.26) a3 = Elrf “Tfi s — Z—g—iHlRiccilz L51?) - 144)
' M
3 2
+ C55 4-c6S +c7S+c8]vg ,

where c5.c6,c7 and c8 are the constants. Since M is

 

n-Einstein, the condition a3 = a; implies
3 2
(5.27) IM (C55 4—c65 4-c754-c8 )vg
. 2
_ . _ 736 _L5 -12)
(M (147 441mRiCCi (212 ‘14“
+ C'S’3+C’S’2+C’S’+C’]V',
5 6 7 8 g '

Where cg,cé,c; and cé are in general expressed in terms

of the dimension of M'. But dim M.= dim M'. Hence

c5 = c5, c6 = c6, c7 = c7 and c8 = c8. In addition we
already have S = S’ = constant. Thus M’ is n-Einstein
if S’ #’736 Since the contact Bochner curvature tensor

8’ of M' vanishes, M’ is of constant w-sectional

I

curvature c = c.
implies c' #”l%% .

 

The restriction 5’ ¢ 736

The converse is also true. Q.E.D.

73

Combining PrOpositions 5.3 and 5.12, we have the
following theorem.

Theorem 5.13. 'Lgp, (M,g) 22g, (M7,g’) be gompact
Sasakian manifoldg with gtructure tensopg_ (m,§,n) “gal
(m’,§',n’) pggpectively. Assume Spec(M,g) = Spec(M’,g'),
then we have

(1), dim M.= dim M’,

(2) for dim M.= dim M' = 5,7,9,ll, M is of

 

constant m-sectiongl,curvature c if and
only if M’ is of constant m—sectional
curvature c' = c,

(3) lpp_ dim M = dim M’ = 13, M is of constant
m-gectional curvature c #V%%% , if and only
ll_ M’ is of constant m-sectionalycurvature

I

c = c.
Since an odd-dimensional sphere S2n+1 with the
standard metric 90 is a compact Sasakian manifold of a
constant m-sectional curvature c = l, we can now give
a partial answer to the question proposed in the Introduction
in the following sense.
Theorem 5.lg, Spec(MR,g) = Spec(Sn,go), under the
assumptions that n = 5,7,9,ll,13 and that MR lg_§

simply connected Sasakian manifold, implies that (MR,g)

is isometric to (Sn,go).

APPENDIX

APPENDIX

SPECTRUM.OF p-FORMS OF SASAKIAN MANIFOLDS

By considering the action of the Laplacian A on
p—forms on a compact orientable Riemannian manifold

(MR,g), we can consider the spectrum of p-forms:

Specp(Mn,g) = {o = >. < l g A2,}; _<_... 1.

0.p 1.13
It is again an interesting problem to investigate
how the spectra {xi Y] reflect the geometry of MR.
I

The asymptotic expansion in this case is
k

‘A- t

(A.1) Zie 1’P = (4‘rrt)-n/2 Z} ai t + 0(tk+l_n/2).

i i=1 '9
The following coefficients are known:
. _ _ n
(A.2) a0,l — n IM vg — n volume of (M ,g)

_ n-6
(A.3) a1,l — -7;— IM Svg,
and
(A 4) a =LJ‘ [2(n-15)]R|2+2(90-n)[Ricci[2
' 2,1 360 M

+ 5(n - 12)sz]vg.
By using the similar technique, we get the following

result:

74

75

Theorem A.l. .lgp (M,g) §pg_ (M’,g’) be compact
Sasakian manifolds. Assume Spec1(M,g) = Spec1(M’,g'),
then we have

(1) dim M.= dim M’,

(2) lpp_ dim M.= dim M’ = l7,l9,21,...,101,103,

M is of constant m-sectional,curvature c,
if and only:if M’ is of constant @-
sectiongl curvature c’ = c.

Corollapy A.g, Specl(MR,g) = Specl(Sn,gO), ppggl_
the assumppions that n = l7,l9,21,...,101,103 and that
MR is a simply connected Sasakian manifold, implies
pp§p_ (Mp,g) is isometric to (Sn,go).

Remark. Tanno [40] proved that Spec1(MR,g) =
Spec1(Sn,gO) implies that (MR,g) is isometric to
(Sn,go) for n = 2.3 or l6,17,18,...,92,93. Hence
our Corollary applies for 95,97,99,101 and 103 dimensional

cases 0

BIBLIOGRAPHY

[l]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[ll]

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M. Berger, P. Gauduchon and E. Mazet, Le spectre
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D.E. Blair, Contact Manifolds in Riemannian Geometry,
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