Holonomy groups of normal bundles

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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.