Content uploaded by Bang-Yen Chen

Author content

All content in this area was uploaded by Bang-Yen Chen on Sep 17, 2014

Content may be subject to copyright.

HOLONOMY GROUPS

OF

NORMAL BUNDLES,

II

M. BARROS

AND

B. Y.

CHEN

1.

Statements

of

theorems

Let

i: N -> M be an

isometric immersion

of a

Riemannian manifold

N

into

a

Riemannian manifold

M. We

denote

by D the

induced connection

on the

normal

bundle

v of N in M.

Then

D is

known

to be a

Riemannian connection

on v

(see,

for

instance,

[3]).

For each point

x of

JV

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

T G

C(X)

the

parallel displacement

in

v along

T

with respect

to D

gives

an

isomorphism

of

the fibre

n~i(x) onto

itself,

where

n is the

projection

of

the normal bundle.

The set of

all such

isomorphisms

of

n~1{x) onto itself forms

a

group, called

the

holonomy group

of

v

with

reference point

x.

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

N

is connected,

all

(restricted) holonomy groups

of

v

are

isomorphic

to

each other.

In

the

first part

of

this series [4],

one of

the authors classifies complete Kaehler

submanifolds

in

compact Hermitian symmetric spaces with trivial (restricted)

holonomy group

of

normal bundle.

In

this part

of the

series

we

shall study

quaternion submanifolds

in an

arbitrary quaternion manifold with trivial (restricted)

holonomy group

of

the normal bundle.

In

particular,

we

shall prove

the

following.

MAIN THEOREM.

Let N

be

a

quaternion submanifold of a quaternion manifold

M.

Then the restricted holonomy group

of

the

normal bundle

is

trivial if and only if both

N

and

M are

Ricciflat.

In this paper

we

shall consider only smooth manifolds

of

dimension greater

than

0.

2.

Preliminaries

A quaternion manifold

is

defined

as a

Riemannian manifold whose holonomy

group

is a

subgroup

of

Sp(m)Sp(i)

=

Sp(m)xSp(l)/{±identity}.

Let M be a 4m-

dimensional quaternion manifold with metric

g.

There exists

a

3-dimensional vector

bundle

V of

tensors

of

type

(1,1)

with local basis

of

almost Hermitian structures

I,J,K

such that (a)

IJ = —JI = K and

(b)

for any

local cross-section

ij/

of

V,

Vxij/

is also

a

cross-section

of V,

where

X is an

arbitrary vector field

in M and

V

the

Levi-

Civita connection

on M. It is

well known that

the

existence

of

such

a

vector bundle

V

on a

Riemannian manifold implies that

it is a

quaternion manifold.

Let

X be a

unit vector

on the

quaternion manifold

M.

Then

X, IX, JX, and KX

form

an

orthonormal frame

in M.

We denote

by

Q(X)

the

4-plane spanned

by

them,

and call

it the

quaternion 4-plane.

Let N be a

quaternion manifold

and i: N

->

M an

Received

8

October,

1979.

[J.

LONDON MATH. SOC.

(2), 22 (1980), 168-174]

HOLONOMY GROUPS

OF

NORMAL BUNDLES,

II 169

isometric immersion from

N

into

M. We

call

N a

quaternion submanifold

of M if

quaternion 4-planes

in N are

carried into quaternion 4-planes

in M by i. For

quaternion manifolds

and

quaternion submanifolds

we

have

the

following

fundamental results.

THEOREM 2.1 ([5], [7]). Every quaternion submanifold of a quaternion manifold

is

totally geodesic.

THEOREM

2.2

(Alekseevskii

[1],

Ishihara

[8]). Any

quaternion manifold

of

dimension

^ 8 is

an Einstein space.

THEOREM 2.3 (Alekseeskii [1], Ishihara [8]).

(a)

When

a

quaternion manifold of

dimension

^ 8 has

nonvanishing scalar curvature,

it is an

irreducible Riemannian

manifold.

(b) When

a

quaternion manifold

of

dimension

^ 8

has zero scalar curvature,

it is

locally

a

Riemannian product

of a

flat quaternion manifold

and an

irreducible

quaternion manifold with

a

vanishing Ricci tensor.

THEOREM

2.4

(Ishihara [8]).

For a

quaternion manifold

of

dimension

^ 8, the

bundle

V

is locally parallelizable if and only if the Ricci tensor vanishes identically, that

is,

in

each coordinate neighbourhood

V

there

is a

canonical basis {I,J,K}

of V

satisfying

V/ = VJ = VK = 0.

Combining the Main Theorem and Theorems 2.1

and

2.3 we have the following.

THEOREM 2.5.

Let N

be

a

quaternion submanifold of a quaternion manifold

M. If

M

is

locally symmetric

and the

restricted holonomy group

of the

normal bundle

is

trivial, then locally

M is a

euclidean space with standard quaternion structure and

N is

a quaternion linear subspace.

3.

Proof

of

Main Theorem and Theorem

2.5

Let

M be a

4m-dimensional quaternion manifold with local almost Hermitian

structures

I, J, and K.

Then, from condition (b),

we see

that there exist three local

1-forms

p, q and r

such that

V*/=

r(X)J-q(X)K,

VXJ= -r(X)I +p(X)K,

VXK=

q(X)I-p(X)J (3.1)

for

any

vector field

X

tangent

to M.

Let

N be a

quaternion submanifold

of

M. Denote

by V and V the

Levi-Civita

connections

of N and M,

respectively. Then,

by

Theorem 2.1,

we

have

VXY = VXY, (3.2)

Vx£

=

Z>^,

(3.3)

170 M.. BARROS AND

B.

Y. CHEN

for vector fields

X, Y

tangent

to

N

and

normal vector field

£,

where

D is the

normal

connection

(on the

normal bundle). Denote

by

R,R'

and RD the

curvature tensors

associated with

V, V and D,

respectively. Then, from

(3.2) and

(3.3),

we

obtain

R(X,Y)Z

=

R'(X,

Y)Z, (3.4)

R(X, Y)£

=

RD(X, Y)Z

(3.5)

for

X,Y,Z

tangent

to

N

and £

normal

to

N.

We first prove

the

following lemmas.

LEMMA

3.1.

Let

N

be a

quaternion submanifold of a quaternion manifold

M

with

flat normal connection (that is,

RD

=

0).

Then for

any

vectors

X,

Y

tangent

to N,

and

Z

tangent

to

M

we

have

R{X,

Y)q>Z

=

<pR(X,

Y)Z,

q>

= I,J

or

K. (3.6)

Proof.

From

(3.1) and

a

straightforward computation

we

have

{VxVy-VyVx-V[x,y]}/

=

a(X, Y)J-P(X,

Y)K, (3.7)

where

a =

1(dr

+

p

A

q)

and /9

=

2(dq

+

r

A

p).

From

(3.7) we

obtain easily that

R{X, Y)IZ

=

IR{X, y)Z + a(X,

Y)JZ~P{X,

Y)KZ, (3.8)

for

X,

Y

tangent

to

N

and Z

tangent

to M.

Now,

let

Z =

^ be a

vector normal

to

N.

Then

by (3.5) and the

assumption

on RD we

have

0 = «(X,Y)Jt-P(XtY)Kt. (3.9)

Since

J£ and K£ are

linearly independent,

(3.9)

implies that

<x(X,

Y)

=

j9(A

r

,

Y) =

0

for any

X, Y

tangent

to N.

Consequently we have

R(X, Y)IZ = IR(X, Y)Z.

Similar

arguments apply

to the

cases

q>

=

J

and

(p

= K.

LEMMA 3.2. Let

N

be a

quaternion submanifold of a quaternion manifold

M

with

flat normal connection. Then

the

Ricci tensor

S' of N

satisfies

0 (3.10)

for

q>

= I,J

or

K

and for vectors

X, Y

tangent

to

N.

Proof

Let

E

lt

.

..,£„,

IE

u

...,IE

n

, J£

l5

...,J£

n

, KE

y

,...,KE

n

be an

orthonormal basis

of the

tangent space

of N.

Then,

by

Lemma 3.1,

we

have

R(X,

Y;

Eh

Ej)

=

R(X,

Y;

<pEh

q>Ej)

(3.11)

for

X,

Y

tangent

to

N

and

i =

1,...,

n,

1(1),...,

I(n), J(l),...,

J(n),

K(l),..., K(n), where

we

put Em =

q>Ea,

a =

1,...,

n

and

R(X,

Y;

Z,

W) = g(R(X, Y)Z, W).

From

HOLONOMY GROUPS

OF

NORMAL BUNDLES,

II 171

(3.11)

we get

0

=

R(X,

Y;

Eh

cpEi) +

R(X,

Y;

ij,Eh «^£,), (3.12)

for

q>,

I/J

e

{/,

J,

K}

with

q> j=

\j/, from which

we

find that

£

R(X,

Y-Ehq>E^

=

Q.

(3.13)

Consequently, from Bianchi's identity

and

Lemma 3.1,

we

obtain

0=

£

= ZR(Et,YiX,Et)+

I

R(Et,q>Xiq>YtEd. (3.14)

i =

1

i = 1

On

the

other hand, from (3.4),

we

find

R(X,Y;Z,W) = R'(X,Y;Z,W)

(3.15)

for

X,

Y,

Z, W

tangent

to

N.

Thus,

by

(3.14)

and

(3.15)

we may

obtain (3.10). This

proves Lemma

3.2.

LEMMA 3.3.

Let

N

be a

quaternion submanifold of a quaternion manifold

M

with

flat normal connection. Then

the

Ricci tensors

S

and S' of M and

N

satisfy

S(X, Y)

=

S'(X, Y)

(3.16)

for vectors

X, Y

tangent

to

N.

Proof.

Let

{flf...,

£m_n,

/£l5..., /£m_n,

J^,

...,J£m-n,

K^,...,

££„,_„}

be an

orthonormal basis

of

the

normal space. Then,

by (3.4) and the

definition

of

Ricci

tensor,

we

have

S'(X, Y) = S(X,

Y)- X

{*(£„ X-

Y,

&) +

*(/&,

X-

Y,

IQ

» =

i

+

R(J£t,X;

Y,JQ

+

R(K£t,X;

Y,

KQ}.

(3.17)

Hence,

we

find that

S'(X, X)

=

S(X,

X)- Y

{K(X,

Q

+

K(X,

IQ

+

K(X, JQ +

K(X

t

K{

t

)},

(3-18)

where

K(X, Y)

denotes

the

sectional curvature

of

the

plane section

X

A

Y.

On the

other hand,

by (3.5) and the

assumption that

RD

=

0, we

have

0

=

R(X,

<pX;

^

q>Q

=

-R(q>Zt,

X;

X,

(pQ-R^t,

X;

X, Q

.

172 M. BARROS AND B. Y. CHEN

Thus

we

have

K{X,£t)

+

K{Xtq>Zt)

=

0,

q>

=

ItJ,K.

(3.19)

In particular,

we

have

K{X,Q

+

K(X,IQ = 0.

(3.20)

Replacing

£,,

by

J£,t, (3.20) implies that

K(X,JQ

+

K(X,KQ = 0.

(3.21)

Substituting (3.20)

and

(3.21) into (3.18)

we

obtain (3.16). This proves Lemma

3.3.

Now

we

return

to the

proof

of

the

Main Theorem.

Let N be a

quaternion

submanifold

of a

quaternion manifold

M. If the

restricted holonomy group

of

normal bundle

of N in

M

is

trivial, then

the

parallel displacement

of

any element

in

v

is independent

of

the

choice

of

path

in

C°(x)

for

any

xin M.

Therefore,

for

any

£en~l(x)

we may

extend

£ to a

local section

in v,

also denoted

by

<!;,

such that

Dx^

= 0

for all

vectors

X

tangent

to

N.

Thus, from

the

definition

of

RD,

we see

that

RD

=

0,

that

is, the

normal connection

is

flat. Since

N is

of

dimension greater than

zero,

M

is of

dimension

^

8.

Thus,

by

Theorem 2.2,

M is

Einsteinian. Consequently,

by Lemma

3.3,

N

is

also Einsteinian. Combining this with Lemmas

3.2 and 3.3 we

see that both

N

and

M

are

Ricci-flat.

Conversely, suppose that both

N

and M are

Ricci flat quaternion manifolds

and

N

is

a

quaternion submanifold

of

M. Then

by

Theorem

2.4 we may

assume that

/,

J

and

K

are

locally parallel. Thus

we

have

R{X, Y)<pZ

=

q>R(X,

Y)Z

(3.22)

for vectors

X,Y,Z

tangent

to M

and

q>

= I,J or K.

Now,

for any

orthonormal

basis

of each normal space

of

N in

M, we

have

t=i

t,X',

Y,KQ]

=0,

(3.23)

for vectors

X, Y

tangent

to N.

On

the

other hand,

by

(3.22)

and the

Bianchi identity,

we

have

R{£t,

X;

Y, Zt) +

R{<p£t,

X;

Y, q>Q

=

R(q>Y,

X;

£„ cpQ (3.24)

for

<p

=

I,JOTK.

Substituting this into (3.23)

we

obtain

£

0

(3.25)

HOLONOMY GROUPS

OF

NORMAL

BUNDLES,

II 173

for vectors

X, Y

tangent

to N and

q>

= I, J or K. Now

replace

£l,...,£m_n

by

HxAi,..., Zm-nl then (3.25) gives

(3.26)

1 = 2

for i/f,

cp

= /, J or X. Combining (3.25) and (3.26) we find that

R(X,

Y; £,

q>Q

= R(X, Y;

<K,

tpM). (3.27)

On the other hand, (3.22) shows that

R(x,

Y;

«K, <PH)

=

-*(*, y;

<K>

"M)

= -R(x,

Y;

Z,

<K)

for

cp =fc

\\J.

Comparing this with (3.27) we obtain

Y;Z,q>Z) = 0 (3.28)

for vectors X, Y tangent to N and t, normal to N. Consequently, for vectors X, Y

tangent to N and

£,,

rj

normal to N, we have

0

= R(X, Y;t

+ ti,

q>Z

+

q>ri)

= 2R(X,

Y;

£,

q>rj)

by virtue of

(4.2).

Since this is true for all vectors

<!;,

rj

normal to N, (3.5) implies that

the normal connection of N in M is flat, from which we may conclude that the

restricted holonomy group of the normal bundle is trivial. This completes the proof

of the Main Theorem.

Let N be a quaternion submanifold of a quaternion manifold M. If the restricted

holonomy group of the normal bundle is trivial, the main theorem implies that M is

Ricci flat; hence M has zero scalar curvature. Thus by Theorem 2.3, M is locally a

Riemannian product of a flat quaternion manifold and an irreducible quaternion

manifold M' with vaniching Ricci tensor. Now suppose that M is locally symmetric;

then M' must be an irreducible, 4/c-dimensional, locally symmetric space. Since such

a space must have nonvanishing Ricci tensor, M is a flat quaternion manifold.

Consequently, M is a euclidean space with standard quaternion structure. Moreover,

from Theorem 2.1, N is a quaternion linear subspace. This proves Theorem 2.5.

References

1. D. V.

Alekseevskii,

"Riemannian spaces with exceptional holonomy

groups",

Funkcional

Anal,

i

Prilozen,

2

(1968),

1-10.

2. D. V.

Alekseevskii,

"Compact quaternion

spaces",

Funkcional

Anal,

i

Prilozen,

2

(1968),

11-20.

3.

B. Y.

Chen,

Geometry

of

Submanifolds

(Dekker,

New

York,

1973).

4. B. Y.

Chen,

"Holonomy groups

of

normal

bundles",

J.

London

Math.

Soc. (2), 18

(1978),

334-338.

5. B. Y.

Chen,

"Totally umbilical submanifolds

of

quaternion-space-forms",

J.

Austral.

Math.

Soc. Ser. A,

26

(1978),

154-162.

6. B. Y.

Chen

and H. S. Lue, "On

normal connection

of

Kaehler

submanifolds",

J.

Math.

Soc.

Japan,

27

(1975),

550-556.

174 HOLONOMY GROUPS OF NORMAL BUNDLES, II

7.

A. Gray, "A note on manifolds whose holonomy group is a subgroup of

Sp(n) •

Sp(l)", Michigan Math.

J., 16 (1969), 125-128.

8. S. Ishihara, "Quaternion Kaehlerian manifolds", J. Differential Geom, 9 (1974), 483-500.

Department of Mathematics,

Princeton University,

Fine Hall,

Princeton,

N.

J. 08544,

U.S.A.

and

Department of Mathematics,

Michigan State University,

Wells Hall,

East Lansing,

Michigan 48824,

U.S.A.