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HOLONOMY GROUPS

OF

NORMAL BUNDLES

BANG-YEN CHEN

1.

Statement

of

theorems

Let

M be a

compact Hermitian symmetric space. Then

M is

simply-connected

and

its

canonical Hermitian structure

is

Kaehlerian. We denote

by V the

canonical

connection

on M.

Let

M

be

a

complete Kaehler manifold (manifolds are assumed

to

be connected

and

differentiable). By

a

Kaehler immersion

of M

into

M

we mean

an

isometric holomorphic immersion from

M

into

M. Let

i:M-+Mbea Kaehler

immersion

of M in M. We

denote

by D the

induced connection

on the

normal

bundle

v of M in M.

Then

D is

known

to be a

Riemannian connection

on v (see,

for instance,

[1]).

For each point

x of M

we denote

by C(x)

the loop space

at x. Let C°(x) be the

subset

of C(x)

consisting

of

loops which

are

homotopic

to

zero.

For

each

x

e C(x)

the parallel displacement

in v

along

x

with respect

to D

gives

an

isomorphism

of

the

fiber n~1(x) onto

itself,

where

n is the

projection

of

the normal bundle. The

set of

all such isomorphisms

of

n~

l(x)

onto itself forms

a

group, called the holonomy group

of

v

with reference point

x (see [4]). The

subgroup

of

the holonomy group

of v

consisting of the parallel displacements arising from all x

e

C°(x) is called the restricted

holonomy group

of v

with reference point

x.

Since

M is

connected,

all

(restricted)

holonomy groups

are

isomorphic

to

each other.

We

denote

by y and y the

Ricci

2-form

of M

and

M

respectively and

i*

the induced map

of i on

differential forms.

In this paper

we

shall obtain

a

necessary

and

sufficient condition

for a

Kaehler

manifold which

can be

immersed

in a

compact Hermitian symmetric space with

trivial restricted holonomy group

on the

normal bundle.

In

fact

we

shall prove

the

following.

THEOREM

1. Let i: M

-*•

M be a

Kaehler immersion from

a

complete Kaehler

manifold

M

into

a

compact Hermitian symmetric space

M.

Then

M is

the Riemannian

product

of i(M) and

another compact Hermitian symmetric space

M',

that

is

J\?

=

i(M)

x M', if

and only

if

the restricted holonomy group

of

the normal bundle

is

trivial.

As

an

application

of

Theorem

1 we

shall prove

the

following.

THEOREM

2. Let i: M

-*•

M be a

Kaehler immersion from

a

complete Kaehler

manifold

M

into

a

compact Hermitian symmetric space

fil. If

the Ricci 2-form

y

of M

is induced from the Ricci 2-form

y of Si,

that

is

i*y

= y,

then

M is

itself

a

compact

Hermitian symmetric space

and M is the

Riemannian product

of i(M) and

another

compact Hermitian symmetric space

M'.

Received

12

May, 1977; revised 13 March, 1978.

Partially supported

by

NSF under Grant MCS 76-06318.

[J.

LONDON MATH.

SOC.

(2), 18 (1978), 334-338]

HOLONOMY GROUPS

OF

NORMAL

BUNDLES

335

The basic facts

on

symmetric spaces we need

in

this paper may

be

found

in [3]

and [4]. For general results

on

normal connection

for

Kaehler submanifolds, see [2].

2.

Basic formulas

Let M be

a

Kaehler manifold of complex dimension n with connection V, complex

structure

J and

metric

g.

Then

the

curvature tensor

R of M is

given

by

R(X,

Y) =

VxVy-VyVx-V[X>

y] for any

tangent vector fields

X and 7. Let

Elt ...,£„, JE1}..., JEn

be an

orthonormal frame

on M. The

Ricci tensor

S(X, Y)

is given

by

S(X,

7) = £

R(Eh

X;

Y,

Et)+ £

R(JEh

X;

Y, JEd,

(2.1)

i=1

1=1

where

R(Eh X;

Y,

E{)

=

g(R(Eh

X)

Y, Et).

The

curvature tensor

R

satisfies

the

following formulas.

R(JX, JY) = R(X, Y), R(X, Y) JZ = JR(X, Y)Z, (2.2)

R(X, Y)Z + R(Y,Z)X + R(Z,X) 7 = 0, (2.3)

R(X, Y; Z, W)

= R(Z,

W;

X, Y) = -R(Y, X;

Z, W).

(2.4)

Let

i: M

-*•

M be a

Kaehler immersion

of M

into

a

complex m-dimensional

Kaehler manifold

M

with connection V and metric g. Then the second fundamental

form

h of M in M is

given

by Vx 7 = Vx 7

+h(X,

Y).

Let JV

be a

normal vector

field of

M in M;

we write

VXN=-AN(X)

+

DXN,

where —

AN(X)

and DXN

denote

the

tangential

and

normal components

of ¥XN.

Then we have g(AN(X),

Y) =

g(h(X,

Y), N), D is called the normal

connection

on

the

normal bundle

v

of M in

M.

We denote by

R1

the curvature tensor associated with £>,

that is, i?x(Z,

Y) =

Dx

DY

—

DY

Dx

—

DlXt

y]. Then the equations

of

Gauss and Ricci

are given respectively

by

R(X, Y;Z, W) = R(X, Y:Z, W)+g{h(X,Z),h{Y, W))-g{h(Y,Z),h(X>

W)),

(2.5)

R{X,

Y;N,N')

= RHX, Y;N,N')-g([AN,AN.](X), 7), (2.6)

where

K is

the curvature tensor

of

M,

X,

Y,

Z, Ware vector fields tangent

to M

and

N,

N'

vector fields normal

to M.

Moreover, we have

A~JN

= JAN and

JAN=-ANJ,

(2.7)

where

J is the

complex structure

on M. The

Ricci 2-form

y of M is

given

by

,

Y) =

S(JX,

7).

We prove

the

following proposition

for

later use.

PROPOSITION.

Let i: M

->

Si be a

Kaehler immersion

of

a Kaehler manifold M

into a

non-negatively curved

Kaehler manifold

fit.

Then the restricted

holonomy

group

of the normal bundle v

is

trivial

if

and only

if

the Ricci 2-forms

of M

and

M

satisfy

i*y

= y.

336 BANG-YEN CHEN

Proof. Let the

complex dimensions

of M and

M

be n and m

respectively. Then

locally there exist orthonormal sections

Nlt

...,iVm_,,,

JN^

..., JNm_n

in

the

normal

bundle

v of M in M.

From

the

definition

of

Ricci tensor

and the

equation

of

Gauss,

we have

S(X,

X) =

S(X, X)

- Y

[R(Na,

X;X,N0)

+

R(JNa,

X\ X,

JNa)

(2.8)

a=l

-Xg(h{EhX),h(EttX»,

where

S is the

Ricci tensor of

iW"

and

Eu

..., E2n is an

orthonormal frame of

M.

From

(2.7)

we

find

Z

i(h(Eh

X), h(Eh X))

=

2 Y

g(Aa\X)y

X),

(2.9)

a =

1

where

Aa =

ylNa. Moreover,

by

(2.2),

(2.3) and (2.4) we

have

R(X, JX; Na,

JNJ =

-R(Na,

X; X,

Na)-R(JNa,

X; X, JNJ.

(2.10)

Combining (2.8),

(2.9)

and

(2.10),

we see

that

the

Ricci 2-forms

y

and

y

satisfy

i*y

= y if

and

only

if

we have

tn—n

m—n

2

S

g(Aa2(X),

X) = Z

&(X>

JX;

Na>

JNa).

(2.11)

cc

=

l a=l

If

the

restricted holonomy group

of

the

normal bundle

v is

trivial, then

the

parallel displacement

of

any

element

in v is

independent

of

the

choice

of

path

in

C0(x)

for

any

x in

M.

Therefore,

for

any N

en~l(x)

we may

extend

N to a

local

section

in

v,

also denoted

by N,

such that

DXN = 0

for all

vector

X

tangent

to M.

By using

(2.6) and (2.7) we get

R(X, JX; N, JN)

=

2g(AN2(X),

X).

This shows that

i*y = y.

Conversely,

if

we

have

i*y

=

y, we

have (2.11). Since

M

is assumed

to be

non-negatively curved,

the

sectional curvature

of

every plane section

in

JM[

is

non-negative. Thus (2.10) implies that

R(X, JX;Na,

JNJ is

non-positive.

Combining this with (2.11)

we get

Aa

= 0, R(X, JX; Na, JNa)

=

0,

a =

1, ...,

m-n.

Substituting these into

(2.6) we

find

Rx(X,JX;N,JN)

=

0

(2.12)

for

any

X

tangent

to M

and

N

normal

to M.

Replacing

X

and

N

by

X+ Y and

N+N'

in

(2.12)

and

applying

(2.2) and

(2.12)

we get

Rx =

0.

Thus

the

restricted

holonomy group

of

the

normal bundle

is

trivial. This completes

the

proof

of

the

proposition.

3.

Proof

of

Theorems

1

and

2

Let

M

be

a

compact Hermitian symmetric space. Then there

is

a

triple (G,

H,

a)

consisting

of a

connected

Lie

group

G,

a

closed subgroup

H of

G and an

involutive

HOLONOMY GROUPS OF NORMAL BUNDLES 337

automorphism a of G such that M = G/H and H lies between Ga and the identity

component of Gff, where

Ga

is the closed subgroup of

G

consisting of

all

elements left

fixed by a. Let g and h be the Lie algebras of G and H, respectively, and let

g =

\)

+ m be the canonical decomposition of g associated with a. Then we have

[I), I)]

c

I),

[h, m] a m and [m, m] c h. In the following, we shall identify m with the

tangent space T0(M) of

A?

at the origin 0. It is known that

?],Z},

(3.1)

for X, f,2e

m.

Let B be the Killing-Cartan form on

g.

Since M is compact, we may

assume that the metric g on M is given by

—

B.

Let i: M -> hi be a Kaehler immersion of a complete Kaehler manifold M on ii?.

Since the metric on A? is G-invariant, for any fixed x e

A?

we may assume that x is

the origin 0 of Af.

Now, suppose that the restricted holonomy group of the normal bundle v is

trivial. Then for any N e n~i

(0)

we may extend N to a local section in v, also denoted

by N, such that DXN = 0 for any vector X tangent to M. Combining this with the

equation of Ricci we get

R(X, Y; N, N') =

-g([AN,

AN-](X),

Y). (3.2)

By using the curvature identities and (2.7) we find

2g{AN\X),

X) = -R{N, X; X,

N)-R(JN,

X; X, JN).

Combining this with (3.1) we get

2g(AN2(X),

X) = B([X,

IV],

[X,

N])

+ B([X, JN], [X, JN]).

Since B is negative-definite on h, it follows that

^ = 0, [X,N] = 0, (3.3)

for all XeT0(M) and Nen'^O). Let n^ = T0(i(MJ) and m2 = n'^O). Then

mx and m2 are two orthogonal subspaces of m such that

m =

in!

+

m2

(direct sum).

Since J\? is Hermitian symmetric, (3.3) implies that M is a complete totally geodesic

submanifold of

A?.

Thus M is also a Hermitian symmetric space and m^ forms a Lie

triple system, that is,[[ml5 mj, mj emt. (3.4)

On the other hand, (3.3) also implies

[ml5rn2] = 0. (3.5)

For any X2, Y2,Z2em2, if we put [[X2,

Y

2

],Z^\

= Wl

+ W2,

for some fluent!

and

W2em2,

then by (3.5) and the Jacobi identity, we find

u

JW,]

=

[[[X2, Y2],Z2l

JWX]

=

0.

If wt it 0, this implies that the holomorphic sectional curvature of

A?

at

W^

vanishes.

Since Af is a compact Hermitian symmetric space, every holomorphic sectional

curvature of M is positive. Hence we get a contradiction. Consequently, we have

proved that m2 also forms a Lie triple system. Let M' be the complete totally geodesic

submanifold in

Al"

with

T0(M')

= m2.

338 HOLONOMY GROUPS

OF

NORMAL BUNDLES

If we

put

*)i

=

[nt!,

mj, f)2 =

[m2,

m2]

g2=f)2+m2s

then (g1} f)l9

at)

and (g2, f)2, cr2)

are two

symmetric subalgebras

of

(g, f),

a).

Moreover

we have

[f)u

f)2]

= 0 and f)i nf)2 =

{0}.

Let f)' =

[in, m],

g' = f)'+m and u' = a|B,.

Then (g',

F)',

a')

also forms

a

symmetric

Lie

algebra.

Let G' be the

connected

Lie

subgroup

of G

whose

Lie

algebra

is g' and set H' =

G'

n #.

Then

M =

G'//*',

f)'

=

t)i+f)2»

9' =

9i+92>

ffi =

^'Im,

and

°"2

=

°"'lm2- From these

we

obtain

a

direct

sum decomposition

of

(g',f)>'). That

is,

(g',l)\

a') =

(g^th,

(T0

+

(g2, f)2,

o2).

The corresponding direct product

of (G\ if',

cr')

is

given

by

(G'}

/f', a') =

(Gl5

Hlt a,) x

(G2,

H2, a2\

where

Gt (i = 1, 2) are the

connected

Lie

subgroups

of G'

whose

Lie

algebras

are

Qh

and Ht = GtnH'.

Consequently

we get a

direct product decomposition

of the

compact Hermitian symmetric space

M = G/H = G'/H': G'/H' = G1/Hl x G2/H2.

From GJHt

= i(M) and G2/if2 = M', we get M = i(M) x

M'. Since A?

is

compact,

M

and M' are

both compact.

Conversely,

if M and M' are

compact Hermitian symmetric spaces

and

fii

=

MxM',

then

the

normal bundle

of M in M is a

trivial vector bundle whose

induced connection

on v is

also trivial. Since

M is

simply-connected,

the

parallel

displacement

in v

with respect

to the

normal connection

is

independent

of the

choice

of

path. Thus

the

holonomy group

and the

restricted holonomy group

of

v

are trivial. This completes

the

proof

of

Theorem

1.

Theorem

2

follows from Theorem

1 and the

proposition immediately.

References

1.

B. Y.

Chen, Geometry of

submanifolds

(M.

Dekker,

New

York, 1973).

2.

B. Y.

Chen

and H. S. Lue, " On

normal connection

of

Kaehler submanifolds

", /.

Math.

Soc.

Japan,

27

(1975), 550-556.

3.

S.

Helgason, Differential

Geometry

and Symmetric Spaces (Academic Press, New York, 1962).

4.

S.

Kobayashi

and K.

Nomizu,

Foundations

of

Differential

Geometry,

vol. I and II

(Interscience

Publ., New York, 1963, 1969).

Department

of

Mathematics,

Michigan State University,

East Lansing, Michigan 48824,

U.S.A.