Totally geodesic submanifolds of symmetric spaces I

Abstract

By applying Lie triple system we classify totally geodesic submanifolds of complex quadrics Q^n=SO(n+2)/SO(2) x SO(n) for n ≥ 2.

Full text

Vol.
44,
No.
4
DUKE
MATHEMATICAL
JOURNAL(C)
December
1977
TOTALLY
GEODESIC
SUBMANIFOLDS
OF
SYMMETRIC
SPACES,
I
BANG-YEN
CHEN
AND
TADASHI
NAGANO
1.
Introduction.
An
isometric
immersion
f:
M
N
of
a
Riemannian
manifold
M
into
another
Riemannian
manifold
N
is
called
totally
geodesic
if
the
geodesics
in
M
are
carried
into
geodesics
in
N.
In
this
paper
we
will
completely
classify
all
the
complete,
connected,
totally
geodesic
submanifolds
of
the
complex
quadratic
hypersurfaces:
Qm
SO(m
+
2)/S0(2)
SO(m),
rn
>
1.
Our
main
result
is,
in
a
simplified
version,
that
every
maximal
connected
totally
geodesic
submanifold
M
of
Qm
is
a
connected
component
of
the
fixed
point
set
of
some
involutive
isometries
of
Qm.
More
concretely
M
is
one
of
the
following
three
spaces:
(1)
Qm-
1,
(2)
the
Riemannian
product
S
v
S
,
rn
p
+
q,
of
two
spheres
of
the
same
curvature,
and
(3)
the
complex
projective
space
of
complex
dimension
n
(if
rn
2n).
Their
immersions
as
totally
geodesic
sub-
manifolds
in
Qm
are
unique
up
to
an
isometry
of
Qm.
Furthermore
we
shall
study
more
details
of
these
spaces.
Among
these
three
spaces
above,
Qm-
is
the
only
complex
submanifold
of
Qm,
the
immersion
of
S S
is
totally
real,
and
the
complex
projective
space
is
neither
a
complex
submanifold
nor
a
totally
real
submanifold
of
Qm.
If
M
is
a
non-maximal,
complete,
connected,
totally
geodesic
submanifold
of
Qm,
then
M
is
contained
in
Qm-
in
some
position,
except
for
the
real
projective
space
of
half
dimension
which
is
contained
in
the
complex
projective
space
mentioned
above
for
an
even
m.
Consequently,
every
connected
totally
geodesic
submanifold
of
dimension
>
of
Qm
is
an
open
submanifold
of
the
common
fixed
point
set
of
a
finite
number
of
involutive
isometries
of
Qm.
The
choice
of
the
space
Qm
was
made
out
of
our
current
interests
beside
its
simplicity.
We
do
not
think
however
that
the
feasibility
is
limited
to
Qm
but
we
hope
that
our
method
will
work
for
other
symmetric
spaces.
In
section
2
we
will
give
examples
of
totally
geodesic
submanifolds
of
Qm
with
detailed
descrip-
tions.
Our
main
results
will
be
proved
in
section
3.
In
the
last
section
we
will
give
some
topological
descriptions
of
those
totally
geodesic
submanifolds.
In
particular,
we
will
prove
that
each
homology
group
Htc(Qm;
Z),
k
<
2m,
is
spanned
by
the
classes
of
totally
geodesic
submanifolds.
And
there
is
a
maxi-
mal
connected
totally
geodesic
submanifold
M
of
Qm
such
that
the
dif-
ferentiable
manifold
Qm
is
the
union
of
the
normal
bundles
to
M
and
to
its
focal
manifold
with
nonzero
vectors
identified
in
some
way.
Received
April
16,
1977.
Revision
received
June
9,
1977.
First
author
partially
supported
by
NSF
under
Grant
MCS
76-06318.
Second
author
partially
supported
by
NSF
under
Grant
MCS
76-06953.
745

746
BANG-YEN
CHEN
AND
TADASHI
NAGANO
The
basic
facts
on
symmetric
spaces
we
need
in
this
paper
may
be
found
in
[4]
and
[5].
2.
The
space
Qm
and
its
totally
geodesic
submanifolds.
We
write
Q
Qm
for
the
Grassman
manifold
G2(E
2
+
m)
and
P
for
the
orient-
ed
2-plane
el/
e2
with
respect
to
the
fixed
orthonormal
basis
(el,
e2
+
m)
for
E
E
+
m.
Some
other
geometric
interpretation
of
Qm
will
be
explained
at
the
end
of
this
section.
Here
we
first
give
a
description
of
the
tangent
space
to
Qm
at
P
which
will
be
used
for
this
and
the
next
sections.
Then
we
will
give
examples
of
totally
geodesic
submanifolds
of
Qm
which
will
turn
out
later
to
exhaust
all
of
the
totally
geodesic
submanifolds
as
our
theorems
assert.
Since
Qm
is
a
submanifold
of
the
vector
space
A2E;
the
tangent
space
to
Qm
at
the
point
P
is
viewed
as
its
vector
subspace
and
as
such
it
is
the
direct
sum
el
/
V
-+-
e2/
V,
where
V
is
the
orthogonal
complement
of
P
in
E.
We
will
give
the
examples
in
this
section
based
on
the
next
lemma.
LEMMA
2.1.
Let
H
be
a
set
of
isometries
of
a
Riemannian
manifold
N.
Then
every
connected
component
M
of
the
fixed
point
set
F(N,
H):
{p
M:
g(p)
p
for
every
g
in
H}
is
a
totally
geodesic
submanifold
of
N.
Proof.
Easy
to
see
if
one
expresses
the
isometrics
in
H
as
linear
transfor-
mations
in
terms
of
a
normal
coordinate
system
at
each
point
of
M.
LEMMA
2.2.
Vectors
v,
w,
x
and
y
in
E
satisfy
the
equation
v/
w
+
x/
y
0
if
and
only
if
either
both
v/
w
and
x/
y
are
zero
or
the
sets
{v,
w}
and
{x,
y}
span
one
and
the
same
2-plane
in
E.
Proof.
Trivial.
From
Lemma
2.2,
we
get
the
following
lemma
immediately.
LEMMA
2.3.
Given
a
linear
involution
sn
orE,
decompose
every
vector
x
in
E
into
x
x+
+
x_
in
such
a
way
that
sn(x+)
x+
and
sR(x_)
-x_.
Then
an
oriented
2-plane
x/k
y
is
fixed
by
sR
induced
on
Qmfrom
s
if
and
only
if
the
plane
x/k
y
is
either
the
plane
x+
/k
y+
or
x_
/k
y_.
The
unoriented
plane
spanned
by
{x,
y}
is
fixed
by
the
induced
Sl
if
and
only
if
the
plane
is
one
of
the
four
2-planes
{x+,
y+}.
This
lemma
is
convenient
to
determine
the
fixed
point
sets
in
the
sequel.
We
introduce
some
notation
to
describe
the
examples
below.
Let
R
be
a
proper
subset
of
the
index
set
{1,
2,
2
+
m}.
Define
an
orthogonal
trans-
formation
s
of
E
by
sn(ei)
ei
or
-e
according
as
R
or
not.
Then
sR
induces
an
involutive
isometry
on
Qm
in
the
natural
fashion,
which
we
denote
by
sn.
We
write
F(N,
g)
for
F(N,
{g}),
when
H
is
a
singleton
{g}.
We
put
En
F(E,
sn).
And
let
S(E)
denote
the
unit
sphere
of
any
metric
vector
space
E.
Example
2.1.
Take
a
subset
R
of
{1,
2,
2
+
n}
which
contains
{1,
2},
say
R
{1,
2,
2
+
n}.
The
fixed
point
set
F(Qm,
sn)
is
the
disjoint
union

TOTALLY
GEODESIC
SUBMANIFOLDS
OF
SYMMETRIC
SPACES,
747
of
Qn
G2(
E
+
n)
and
G2(E
TM
n).
Qn
contains
P
and
its
tangent
space
at
P
is
/
ER
+
e/
En,
where
En
F(E,
sn).
If
n
rn
1,
Q,
may
be
viewed
as
the
set
of
all
the
2-planes
which
are
orthogonal
to
era.
Example
2.2.
Suppose
a
set
R
of
{1,
2,
2
+
m}
does
not
contain
but
2,
says
R
{2,
3,
2
+
p},
2
_-<
p
-<
m.
Let
So
denote
the
involution
of
which
carries
every
point
x/
y
into
y/
x,
the
same
plane
with
the
opposite
orientation.
Then
F(Q,
So
sn)
is
locally
the
Riemannian
product
S
p
S
q,
p
+
q
m.
The
immersion
f:
S
v
S
--*
Q,,
carries
(x,
y)
S(En)
S(E)
into
x/
y,
where
E
is
the
orthogonal
complement
of
En
in
EE
+
m.
f
is
not
an
imbed-
ding,
sincef(-x,
-y)
f(x,
y).
Later
we
will
show
that
the
immersed
S
has
the
same
sectional
curvature
as
immersed
S
if
both
p
and
q
>-
2.
If
p
m,
f(S
p)
may
be
viewed
as
the
set
of
all
the
2-planes
which
contain
el.
Example
2.3.
Suppose
m
is
even.
Then
E
+
m
has
a
complex
structure
j
defined
by
j(ezk-
1)
ezk
and
j(e)
-e2_
for
-<
K
-<
-(2
+
m).Then
j
gives
rise
to
an
involution
s(j)
of
Q,,
which
carries
x/
y
intojx/jy.
Moreover
F(Qm,
s(j))
{x
/
jx:
x
S(E)}
is
the
complex
projective
space
of
real
dimension
m.
Remark
2.1.
If
E
+
m
denotes
the
tangent
space
to
a
Riemannian
manifold
and
the
sectional
curvature
is
interpreted
as
a
function
on
Q,,,
then
the
holo-
morphic
bisectional
curvature
of
Goldberg-Kobayashi
is
its
restriction
to
F(Qm,
so)).
Example
2.4.
The
intersection
of
two
totally
geodesic
submanifolds
is
of
course
totally
geodesic.
For
instance,
let
rn
be
even
and
R
be
the
set
of
odd
integers
in
{1,
2,
2
+
m}.
Then
F(Qm,
{So
sn,
so)})
F(Qm,
So
sn)
V)
F(Qm,
is
another
totally
geodesic
submanifold.
This
is
a
real
projective
space,
which
is
the
image
of
the
totally
geodesic
immersion
f:
S(En)
Qm
defined
byf(x)
x
ix.
LEMMA
2.4.
Every
totally
geodesic
submanifold
of
Qm-
c
Qm
is
totally
geodesic
in
Qm.
Proof.
The
composition
of
two
totally
geodesic
maps
is
obviously
totally
geodesic.
In
order
to
study
more
about
the
totally
geodesic
submanifolds
in
the
exam-
ples,
we
will
calculate
the
curvature,
which
will
also
be
used
later.
We
keep
the
previous
notation.
The
tangent
space
to
Qm
at
P
e/
e
is
e/
V
+
e/
V
which
is
identified
with
mQ.
The
members
X
of
mQ
may
also
be
viewed
as
skew-
symmetric
linear
transformation
of
E
+
m
which
carry
e
into
Xe
in
V,
a
1,
2.
The
tangent
vector
corresponding
to
X
at
P
is
then
written
as
XP
(XeO

748
BANG-YEN
CHEN
AND
TADASHI
NAGANO
e2
+
el/
(Xe2).
The
bracket
product
of
XP
and
YP
for
X,
Y
mQ
as
members
of
mQ
is
a
member
of
which
is
the
Lie
algebra
go(2)
x
go(m)
of
skew-
symmetric
transformations
of
E
leaving
invariant
the
2-plane
el
/
ez
(and
so
V
too).
LEMMA
2.5.
Let
al
(Xel,
Yez)
(Yel,
Xez).
Then
[XP
YP]
a
l2e
/
e.
(Xe
l)
/
Ye
l)
(Xez)
/
Ye).
Proof.
Trivial.
The
coefficient
all
may
be
regarded
as
a
Ko-invariant
skew-symmetric
bilin-
ear
form
on
mo,
which
restricts
to
a
K-invariant
form
on
mt
for
any
totally
geodesic
submanifold
M
of
Qm
when
(X,
y)
is
orthonormal,
the
sectional
curva-
ture
of
the
plane
X
A
y
is
II[xe,
YP]ll
([43,
p.
206).
From
this
we
immediately
obtain
the
next
two
lemmas.
LEMMA
2.6.
The
curvatures
of
the
immersed
spheres
S
p
and
S
q
in
Example
2.2
are
equal
to
each
other
if
p
and
q
>
1.
LEMMA
2.7.
The
real
projective
space
in
Example
2.4
has
the
sectional
curvature
equal
to
-
I]e
A
e2[[.
The
symmetric
space
Qm
is
Hermitian.
The
complex
structure
J
restricted
to
the
tangent
space
TpQm
is
given
by
the
next
formula.
(2.1)
J(XP)
(Xez)
/
el
+
el/
(Xe
l).
Thus,
the
coefficient
all
in
Lemma
2.5
has
the
following
meaning.
(2.2)
all
(J(XP),
YP).
A
submanifold
M
of
a
complex
manifold
with
the
complex
structure
J
is
called
totally
real
if
and
only
if
the
tangent
space
TvM
is
orthogonal
to
J(TM)
(with
intersection
{0})
for
every
point
p.
LEMMA
2.8.
A
totally
geodesic
submanifold
M
of
Qm
is
totally
real
if
and
only
if
[mt,
mt]
is
contained
in
go(m).
Proof.
By
(2.2),
M
is
totally
real
if
and
only
if
all
0
for
every
pair
of
vectors
X,
Y
in
mt.
LEMMA
2.9.
The
immersed
SS
in
Example
2.2
is
totally
real,
while
the
complex
projective
space
in
Example
2.3
is
not.
Proof.
Follows
from
Lemma
2.8.
Om
can
be
viewed
as
a
homogenous
manifold
S0(2
+
m)/SO(2)
SO(m).
S0(2
+
m)
is
locally
isomorphic
with
the
whole
group
of
the
isometries
of
Qm;
S0(2
+
m)
is
not
necessary
effective.

TOTALLY
GEODESIC
SUBMANIFOLDS
OF
SYMMETRIC
SPACES,
749
We
give
another
interpretation
of
Qm.
Since
Qm
is
a
Hermitian
symmetric
space,
Qm
is
algebraic
and
indeed
a
complex
hypersurface
of
the
complex
pro-
jective
space
P(C
2
/
m)
of
complex
dimension
+
m.
Its
equation
is
2+m
o
in
terms
of
the
homogeneous
coordinates
(z)
e
+
m.
The
action
of
S
U(2
-+-
m)
of
C
+
m
as
a
unitary
group
is
induced
on
P(C
/
m)
and
its
members
which
leave
the
quadratic
hypersurface
above
form
a
subgroup
which
is
easily
seen
to
be
S0(2
+
m)
with
the
isotropy
subgroup
SO(2)
SO(m).
A
direct
correspondence
between
the
two
spaces
may
be
obtained
by
assigning
z
x
+
iy
C
/
m
to
an
orthonormal
basis
of
the
oriented
2-plane
x/k
y.
With
this
interpretation,
one
easily
recovers
the
totally
geodesic
submani-
folds
in
the
Examples;
for
instance,
Om--1
{[’]:
Z2
+
m
0}
and
So([Z])
[].
3.
The
main
theorems
and
their
proofs.
In
2
we
constructed
some
examples
of
totally
geodesic
submanifolds
of
Qm
S0(2
+
m)/SO(2)
SO(m).
The
following
theorems
assert
that
such
ex-
amples
exhaust
all
the
totally
geodesic
submanifolds
of
Qm
virtually.
In
this
section
we
make
the
following
Assumption.
M
is
a
complete,
connected,
Riemannian
manifold
of
dimen-
sion
>
and
Qm
is
the
complex
quadratic
hypersurface
S0(2
+
m)/SO(2)
SO(m)
of
complex
dimension
m.
TIaEOREM
3.1.
(Characterization).
M
is
a
totally
geodesic
submanifold
of
Om
if
and
only
if
M
is
a
connected
component
of
the
fixed
point
set
of
a
finite
set
of
involutions
of
Qm.
THEOREM
3.2.
(Uniqueness).
The
image
of
an
isometric
totally
geodesic
imbedding
of
M
into
Qm
is
unique
up
to
an
isometry
of
Om.
THEOREM
3.3.
(Primary
classification).
If
M
is
a
maximal
totally
geodesic
submanifold
of
Qm,
M
is
one
of
the
following
three
spaces:
(2)
a
local
Riemannian
product
of
two
spheres
S
and
S
,
p
+
q
m,
and
(3)
the
complex
projective
space
P(C
+
)
of
complex
dimension
n,
2n
m.
THEOREM
3.4.
(Secondary
classification).
If
M
is
a
non-maximal,
totally
geodesic
submanifold
of
Om,
M
is
either
contained
in
Om-
in
an
appropriate
position
in
Qm,
or
the
real
projective
space
P(R
/
)
of
real
dimension
n,
2n
m,
which
is
the
intersection
of
P(C
+
1)
in
(3)
of
Theorem
3.3
and
the
local
product
space
in
(2)with
p
q
n.
THEOREM
3.5.
A
maximal
totally
geodesic
submanifold
M
of
Qm
is
a
com-
plex
submanifold
if
and
only
if
M
Qm-
1.

750
BANG-YEN
CHEN
AND
TADASHI
NAGANO
THEOREM
3.6..
A
maximal,
totally
geodesic
submanifold
M
of
Q,,
is
totally
real
if
and
only
if
M
is
the
local
product
in
(2)of
Theorem
3.3.
THEOREM
3.7.
All
the
spheres
S
p,
2
<-_
p
<-_
m
in
(2)
of
Theorem
3.3
have
the
same
sectional
curvature
which
is
twice
of
that
of
the
real
projective
space
in
Theorem
3.4.
From
Theorems
3.3,
3.4
and
3.7
we
have
the
following
corollaries.
COROLLARY
3.1.
A
maximal
connected
totally
geodesic
submanifold
of
the
Riemannian
product
S
x
S
q,
p
and
q
>
1,
is
either
one
of
S
and
S
or
a
sphere
Sofa
different
curvature
which
is
the
diagonal
set
of
S
x
S
,
p
q.
COROLLARY
3.2.
A
maximal
connected
totally
geodesic
submanifold
of
the
complex
projective
space
P(C
+
)
is
either
P(C
)
in
an
appropriate
position
or
the
real
projective
space
P(R
+
).
The
idea
of
the
proof
of
Theorems
3.2,
3.3
and
3.4
is
to
investigate
the
tangent
space
TvM
to
M
at
a
point
as
Kt-module
or
rather
its
position
in
TuQm
so
as
to
see
that
one
can
carry
TM
into
a
tangent
space
to
one
of
the
spaces
in
2
with
some
isometry
g
of
Qm.
Then
g
will
map
the
whole
space
M
onto
that
space.
We
shall
keep
the
previous
notations.
Proof
of
Theorems
3.2-3.4.
Because
M
is
a
totally
geodesic
submanifold,
M
is
a
symmetric
space.
Since
Go
is
transitive
on
Q,,,
we
may
assume
that
M
contains
the
point
P
el/
ez.
We
have
a
Lie
algebra
monomorphism
h.
gM-">
gQ,
by
which
we
may
regard
gM
as
a
subalgebra
of
ga
o(2
+
m).
Thus,
we
have
fM
C
t.
Moreover,
o(2
+
m)
can
be
regarded
as
skew-symmetric
linear
endo-
morphisms
of
E
+
".
We
recall
that
fa
o(2)
+
o(m),
where
o(2)
and
o(m)
are
taken
as
the
Lie
algebra
of
O(EI,)
and
O(E,...,
+
,,).
The
restriction
h.
mM--->
ma
is
identified
with
the
inclusion
of
TpM
into
T,Qm.
We
denote
by
me
the
sub-
space
{Xe
X
raM}
of
the
space
E,...,
+
m
for
e
E,z,
where
X
acts
on
e
as
a
skew-symmetric
linear
endomorphism
of
E
+
to
give
Xe.
We
need
the
following
lemmas.
LEMMA
3.1.
If
t
C
O(m)
and
t
is
irreducible
on
mt,
then
either
(1)
me
mez
or
(2)
me
is
orthogonal
to
me.
Proof.
The
assumption
implies
te
0,
a
1,
2;
that
is,
e
is
fixed
by
ft.
Thus,
ft
leaves
me
invariant,
and
the
linear
map:
m
me
given
by
X
-
Xe
is
ft-equivariant.
Since
fM
is
irreducible
on
m,
the
map
is
either
0
or
a
kt-
isomorphism.
Now
suppose
the
conclusion
of
the
lemma
is
false.
Then
both
me1
and
me2
are
ft-isomorphic
with
m
and
their
intersection
reel
f’l
me2
{0}.

TOTALLY
GEODESIC
SUBMANIFOLDS
OF
SYMMETRIC
SPACES,
751
In
particular,
if
X
and
Y
are
linearly
independent
vectors
in
m,
then
so
are
Xel
and
Ye..
Since
t
leaves
me2
invariant,
the
vector
[XP,
YP]Ze2
must
lie
in
mez
for
any
X,
Y
and
X
in
m.
This
vector
equals
(Sel/
Yel)Zez
+
(Sez/
Yez)Ze
by
Lemma
2.5
and
the
assumption
on
c
(m).
Since
the
second
term
lies
obviously
in
me.,
so
does
the
first
term
which
is
equal
to
(Eel,
Zeg)Xel
(Xel,
Zez)Eel.
Since
dim
M
>
1,
there
exist
two
linearly
inde-
pendent
vectors
X
and
Y
in
m.
Thus,
Xel
and
Ee
are
linearly
independent
vectors
in
me1.
Therefore
we
have
(Ye,
Zez)
0
and
(Xel,
Zeg)
0
for
every
Z
in
m
whenever
X
and
Y
are
linearly
independent.
Thus,
me1
is
orthogonal
to
me.
This
gives
a
contradiction.
LEMMA
3.2.
In
the
case
(1)
of
Lemma
3.1,
M
satisfies
me
0
after
mov-
ing
M
in
Qm
with
some
rotation
in
SO(2)
c
KQ.
Proof.
Consider
the
bilinear
form
(Sel,
Ye2)
on
m,
X
and
Y
in
m.
Since
the
assumption
C
(m)
implies
a19.
0
for
X,
Y
in
Lemma
2.5,
this
bilinear
form
is
symmetric.
Moreover
this
form
is
t-invariant.
Thus,
it
is
a
constant
multiple
of
the
inner
product
(X,
Y)
by
Schur’s
Lemma
and
so
there
is
a
positive
con-
stant
c
such
that
(Xel,
Ye2)
c
(Sel,
Eel)
for
X
and
Y
in
m.
This
yields
(Xel,
c
Yel
Ye2)
0.
Thus,
by
the
assumption
we
find
Ye2
c
Yel
for
every
Y
in
m.
Therefore
Xel/k
e
+
el/k
Xe
Xel/k
(e.
c
el).
Hence,
if
we
ro-
tate
M
with
an
orthogonal
transformation
A
from
SO(2)
C
KQ
such
that
A(e2)
is
parallel
to
e2
cel,
then
we
have
me
0
for
this
new
submanifold.
LEMMA
3.3.
In
the
case
of
Lemma
3.2,
we
further
assume
that
both
me1
and
me.
are
different
from
{0}.
Then
there
is
an
orthogonal
transformation
B
in
SO(m)
such
that
the
tangent
space
TpMis
{(Sel)/
e
+
el/
B(XeOIX
m}.
Proof.
Since
the
map
h,
M--->
T,M
{Xe
/
e2
+
e
/
Xe2lX
m}
is
isometric,
we
have
(X,
Y)
(Xel,
Ye)
+
(Xe2,
Ye2)
for
X
and
Y
in
m.
The
two
terms
in
the
right
hand
side
are
symmetric
bilinear
forms
on
m,
and
they
are
K-invariant.
Since
Kt
is
irreducible,
by
Schur’s
lemma,
there
exist
two
positive
numbers
c
and
s
such
that
c
+
s
1,
(Xea,
Eel)=
c2(X,
ID,
and
(Xe,
Ye2)=
sZ(X,
Y)
for
X
and
Y
in
m.
We
want
to
show
c
s.
This
is
true
basically
for
the
reason
that
if
h
is
a
homomorphism
of
a
simple
Lie
algebra
into
another
Lie
algebra,
then
its
scalar
multiple
ch
cannot
be
a
homomorphism
for
c
0,
1.
More
tech-
nically,
the
equality
c
s
is
shown
by
calculating
[IX,
Y],
Z]e,
a
1,
2,
for
X,
Y
and
Z
in
rn
in
two
ways.
Assuming
X
is
orthogonal
to
Y,
we
have
the
length
liE[x,
rq,
z]elll
cllE[X,
rq,
Z]ll
on
one
hand.
On
the
other
hand,
this
is
equal
to
z) llsll
/
(s,
z3 ll l )
by
Lemma
2.5.
Comparing
this
with
the
analog
for
e2,
we
obtain
cs
cZs
which
implies
c
s
by
cs
O.
There-
fore
it
is
now
clear
that
there
is
an
orthogonal
transformation
B
in
SO(m)
which

752
BANG-YEN
CHEN
AND
TADASHI
NAGANO
restricts
to
a
KM-equivariant
isometry"
me1
me2.
This
factors
the
map"
rn
me2
into
the
composition"
rn
--
me
--.
mez.
LEMMA
3.4.
If
fM
is
not
contained
in
o(m)
but
fM
is
irreducible
on
m,
then
M
is
Hermitian
and
me
me..
Moreover,
the
tangent
space
TpM
is
either
(1)
e
/
me
+
me1/
e,
or
(2)
{(Xe0/
e
+
e/
(jX)e2IX
m},
where
j
is
one
of
the
two
invariant
complex
structures:
rn
--
rn
of
m.
Proof.
The
homomorphism:
0(2)
+
o(m)
o(2),
given
by
the
composition
of
h.
and
the
projection,
is
not
zero
by
the
assump-
tion,
thus
it
is
surjective.
Since
Kt
is
compact,
it
follows
from
Schur’s
lemma
that
the
center
of
Ku
has
dimension
one.
Thus,
M
is
Hermitian.
Let
j
be
the
one
of
the
fu-invariant
complex
structure
of
m.
We
find
it
is
convenient
to
change
the
notation
slightly
for
the
image
tiM)
of
an
isometric
totally
geodesic
immer-
sion
f
of
universal
covering
space
of
M,
which
we
denote
by
the
space
M.
We
have
homomorphism
h
Gu
Go
which
is
a
local
monomorphism.
In
particu-
lar
h(j)
4
and
h(j)
-1
at
least
in
h.(m).
The
same
theorem
also
gives
that
h(j)
belongs
to
Ko
SO(2)
SO(m)
modulo
its
center.
We
write
(A,
B)
for
h(j),
where
A
SO(2)
{1}
and
B
{1}
SO(m).
Then
A
4
1,
and
B
4
1.
Since
h(j)
=
-1
on
h,(m),
we
have
either
(a)
a
=
-1
and
B
or
(b)
A
and
B
z
-1.
Since
j
centralizes
K,
(A,
B)
centralizes
h(Kt).
Since
(A,
1)
centralizes
Ko,
we
conclude
that
(1,
B)
centralizes
h(Kt).
Case
(a):
A
-1
and
B
1.
If
B
4:
_+
1,
h,(m)
+
Bh,(m)
will
be
a
direct
sum
by
the
irreducibility.
Thus,
by
rotatingf(M)
with
rotation
in
SO(m)
which
carries
every
:
in
h,(m)
into
(:
+
B)/X/,
we
have
B
for
the
new
immer-
sion
which
is
the
composition
of
this
rotation
with
the
given
f.
Thus,
by
an
appropriate
choice
ofj
between
j
and
-j,
we
see
that
h(j)
is
the
invariant
com-
plex
structure
J
A
at
least
on
h,(m);
in
other
words,
f
is
holomorphic.
By
(2.1)
we
have
(j;)e
-Xe
for
X
m.
Hence
mea
me.
Since
the
tangent
space
h,(m)
admits
no
other
h(Kt)-invariant
complex
structure,
it
follows
that
h,(m)
is
the
direct
sum
of
mel/
e
and
el/
me2.
Thus,
we
obtain
(1)
in
the
lemma.
Case
(b):
A
s=
and
B
-1.
Since
(A,
1)
(___
1,
1)
leaves
h,(m)
in-
variant,
(1,
B)
also
leaves
h,(m)
invariant.
Hence,
each
me
is
also
(1,
B)-in-
variant,
ct
1,
2.
Thus,
we
may
assume
B
h(j)
on
h,(m).
As
in
Lemma
3.1
we
have
therefore
either
me
me.
or
me
orthogonal
to
me.
But
the
later
case
cannot
occur,
since,
by
the
assumption,
a
a
in
(2.2)
is
not
zero
on
m
A
m.
Thus,
we
have
me1
me.
and
these
are
Kt-isomorphic
with
m.
Again
by
the
fact
al
0
we
see
from
Schur’s
lemma
that
the
invariant
skew-symmetric
bilin-
ear
form
(Xel,
Ye),
X
and
Y
in
m,
must
equal
c(x,
j
Y)
for
some
nonzero
con-
stant
c.
It
remains
to
show
that
c
(or
1).
The
isotropy
group
Kt
is
locally
the
direct
product
of
a
circle
group
SO(2)
and
a
semisimple
normal
subgroup
St.
The
space
is
the
direct
sum
of
an
St-invariant
subspace
u
and
ju.
Since
we

TOTALLY
GEODESIC
SUBMANIFOLDS
OF
SYMMETRIC
SPACES,
753
have
(u,
ju)
0,
it
follows
that
(uel,
ue)
0.
We
see
c
by
the
arguments
of
Lemma
3.3,
which
applies
because
of
the
fact
h(St)
c
SO(2).
This
proves
the
lemma.
LEMMA
3.5.
If
M
is
reducible,
then
m
is
the
direct
sum
of
two
irreducible
Kspaces
m,
a
1,
2,
and
after
some
rotation,
me
me
and
mex
is
or-
thogonal
to
me.
Proof.
Since
the
symmetric
space
Qt
has
rank
2,
M
is
then
necessarily
a
local
Riemannian
product
of
exactly
two
irreducible
symmetric
spaces
M1
and
M
with
dim
M1
and
dim
M
_->
1.
We
may
assume
dim
M
>
since
there
is
no
problem
in
the
case
dim
M
2.
Since
M
M
x
M
locally,
the
group
Gt
is
a
local
product
G1
G.
and
similarly
for
ft
1
+
2
and
rn
ml
+
m2.
More-
over,
we
know
that
(a)
fl
is
irreducible
on
mle,
and
trivial
on
me,,
a
1,
2.
(b)
f
is
irreducible
on
me,
and
trivial
on
mien.
Since
[m,
m]
0,
we
also
have
(c)
(Xel)/
(Yel)
+
(Xe)/
(Ye)
0
and
(d)
(Xel,
Ye)
(Yel,
Xe)
for
every
pair
(X,
Y)
in
ml
x
m.
We
may
assume
mle
#
O.
Suppose
Xe/
Ye
0
for
some
pair
(X,
Y).
Then
{Xel,
Yel}
spans
the
same
2-dimensional
space
V
as
{Xe,
Ye}.
Thus,
we
have
Yel
cYe
for
some
scalar
c,
since
these
are
the
only
K-invariant
vectors
in
V
by
(a).
By
(b)
and
Schur’s
lemma,
Ye
c
Ye2
is
also
true
for
every
Y
in
m
for
some
c,
whether
or
not
dim
M2
>
1.
Therefore
we
can
always
make
me
0
with
some
rotation
as
in
the
proof
of
Lemma
3.2.
If
Xel/
Y
el
0
for
every
pair
(X,
Y)
on
the
contrary,
we
still
have
mel
O,
since
mlel
does
not
contain
a
nonzero
Kl-invariant
vector
by
(a).
From
this
together
with
(c),
we
obtain
me
0
for
a
similar
reason,
noting
that
m2e2
4:
O.
Finally
(d)
yields
that
me
mlel
is
orthogonal
to
me
me.
Now,
we
return
to
the
proof
of
the
theorems.
The
submanifold
M
is
indeed
congruent
with
one
of
the
spaces
in
Examples
2.1
through
2.4
as
follows.
First
we
assume
that
M
is
irreducible.
If
ft
c
)(m),
then
either
(1)
me1
me
or
(2)
me1
is
orthogonal
to
me
by
lemma
3.1.
In
the
case
(1),
M
is
congruent
with
the
sphere
S
’
in
Example
2.2,
for
<
p
<
rn
by
Lemma
3.2,
as
one
sees
by
moving
M
with
a
rotation
of
E
(and
hence
of
Q)
which
fixes
P
el/k
e2
and
carries
the
space
spanned
by
el
and
me1
in
Lemma
3.2
into
En
in
Example
2.2.
In
the
case
(2),
M
is
the
complex
projective
space
P(O’
+
1),
2p
dim
me1
(see
Example
2.3).
Now,
suppose
M
is
reducible.
Then
Lemma
3.5
shows
that
M
is
a
space
in
Example
2.2.
This
completes
the
proof
of
Theorems
3.2,
3.3
and
3.4.
Theorems
3.1,
3.5
and
3.6
then
follows
from
Lemma
2.1,
Theorems
3.2,
3.3
and
3.4,
and
the
constructions
of
Examples
2.1,
2.2,
2.3
and
2.4.
From
Lemmas
2.6,
2.7
and
the
proof
of
Theorems
3.2,
3.3
and
3.4,
we
get
Theorem
3.7.
4.
Topological
observations.
The
totally
geodesic
submanifolds
of
Q,
are
important
parts
of
Qm
topologi-
cally
too,
as
we
explain
briefly
without
proofs
in
this
section.
Because
of
the
specialities
of
Q,,,
there
are
several
methods
of
varying
effectiveness;
to
use
(a)

754
BANG-YEN
CHEN
AND
TADASHI
NAGANO
the
harmonic
forms,
(b)
certain
exact
sequence
in
algebraic
topology
such
as
Gysin’s
and
(c)
theorems
in
algebraic
geometry
such
as
the
Lefschetz
theorem.
Since
Qm
is
a
symmetric
space,
the
harmonic
forms
are
exactly
the
S0(2
+
m)-
invariant
forms
by
1.
Cartan’s
theorem
[3].
To
describe
them,
we
thus
take
their
values
at
a
point,
say
el
A
e2.
The
tangent
space
there
has
the
orthonormal
basis
consisting
of
el
A
ee
and
e2
A
e,
k
3,
2
+
m.
Let
ae,
fie
be
the
dual
basis.
Put
and
E
(
aa
A
A
az
+
m
+
/3a
A
A
flz
+
m,
if
m
is
even,
0,
if
m
is
odd,
in
the
exterior
algebra
over
the
cotangent
space.
It
is
true
that
these
f
and
E
generate
the
algebra
of
all
the
invariant
forms,
which
is
isomorphic
with
the
cohomology
ring
H*(Qm;
IR),
not
just
as
the
additive
group.
And
the
cohomolo-
gy
ring
H*(Qm;
Z)
is
well-known
(see,
e.g.
[6]).
Since
Qm
is
homogeneous
Kaehlerian,
H*(Qm;
Z)
has
no
torsion.
The
above
gives
a
complete
(up
to
non-
zero
constant
multiples)
description
of
the
cohomology
ring
of
Qm.
Now
let
M
be
a
maximal
connected
totally
geodesic
submanifold
of
Qm;
we
exclude,
how-
ever,
S
p
S
q,
p
+
q
m
and
0
<
p
<
m,
since
S
p
is
contained
in
S
m
and
so
S
S
q
is
homologous
to
zero
in
Qm
obviously.
Then
our
result
in
the
preceding
sections
shows
that
M
is
S
m,
Qm-
1,
or
P(C
n
+
1)
with
2n
m
and
that
Gu
acts
transively
on
the
unit
normal
vector
bundle
S
to
M.
Therefore,
all
the
other
orbits
have
codimension
one
except
the
other;
exceptional
orbit
N
([2],
[7]).
N
is
also
totally
geodesic,
since
Gt
is
transitive
on
S
which
may
be
viewed
as
the
unit
normal
bundle
to
N.
In
terms
of
Riemannian
geometry,
N
is
characterized
as
the
focal
manifold of
M
(that
is
the
complement
of
the
maximal
dif-
feomorphic
image
of
an
open
neighborhood
of
the
zero
section
iri
the
normal
bundle
to
N
under
the
exponential
mapping),
since
codim
N
>
1.
Again
our
classification
shows
that
G
is
SO(1
+
m),
SO(1
+
m)
or
U(1
+
n)
according
as
M
S
m,
Qm-
or
P(C
+
1)
for
2n
m
and
that
N
is
accordingly
Qm-
1,
am
or
P(C"
/
1)
(in
a
different
position)
respectively.
The
complement
of
S
in
Qm
has
two
connected
components,
which
are
the
normal
bundles
to
the
excep-
tional
orbits
M
and
N.
The
normal
bundle
to
M
is
Gt-isomorphic
with
the
tangent
bundle
to
M
unless
M
Qm-
1.
When
M
Qm-
1,
the
normal
bundle
is
the
hyperplane
section
bundle
(that
is
the
first
Chern
class
of
the
normal
bundle
is
given
by
the
fundamental
class).
These
facts,
together
with
Lefschetz
theorem
[1]
gives
the
isomorphisms"
He(Qm;
Z)
Hc(Qm-
1;
Z),
for
k
:
m
or
2m,
Hm(Sm;
Z)
Hm(Qm-
1;
Z)
Hm(Qm;
Z),

TOTALLY
GEODESIC
SUBMANIFOLDS
OF
SYMMETRIC
SPACES,
755
induced
by
the
inclusion:
a
m
to
Qm_
-->
Om,
and
Hm(Qm;
Z)
Hm(Om_
9,;
Z),
if
rn
is
odd.
When
rn
is
odd,
S
m
is
thus
homologous
to
zero
in
Qm
and
S
m
which
is
not
homologous
to
0
in
Qm-
is
contained
in
S
m
and
hence
homologous
to
0
in
Qm
too.
As
to
P(C
/
1)
for
2n
m,
the
inclusion
map
induces
the
isomorphisms:
Hk(P(C,+
1);
Z)
Hk(Qm;
Z),
for
k
<
m,
and
the
inclusion:
M
tO
N--
Qm
induces
the
isomorphism:
Hm(P(C
+
);
Z)
@
Hm(P(C
+
);
Z)
nm(Om;
Z).
f
is
the
pullback
of
the
universal
characteristic
class
C
for
SO(2)
from
Qoo.
and
E
are
the
only
primitive
classes
in
H*(Qm;
R)
(for
the
definition
of
primitive
classes,
see
[8,
p.75]).
Moreover,
f
and
E
are
the
Poincar6
duals
of
Qm-
and
S
m
(up
to
nonzero
constant
multiples)
respectively.
The
presence
of
E
shows
the
existence
of
nontrivial
holomorphic
m-forms
on
Qm;
there
are
no
others.
We
summarize
some
important
facts
to
state
the
next
theorems:
THEOREM
4.1.
There
is
a
maximal
connected
totally
geodesic
submanifold
M
of
Qm
such
that
the
differentiable
manifold
Qm
is
the
union
of
the
normal
bundles
to
M
and
to
its
focal
manifold
N
with
the
nonzero
vectors
identified
in
some
way.
THEOREM
4.2.
Each
homology
group
Hc(Qm;
Z),
k
<
dim
Qm
2m,
is
spanned
by
the
classes
of
totally
geodesic
submanifolds.
And
the
cohomology
ring
H*(Qm;
Z)
is
generated
by
the
Poincar
duals
of
totally
geodesic
submani-
folds.
REFERENCES
1.
A.
ANDREOTTI
AND
T.
FRANKEL,
The
Lefschetz
theorem
on
hyperplane
sections,
Ann.
of
Math.
(2)69(1959),
713-717.
2.
G.
BREDON,
Compact
Transformation
Groups,
Academic
Press,
New
York,
1972.
3.
].
CARTAN,
La
topologie
des
espaces
homogdnes
clos,
M6m.
Sem.
Anal.
Vect.
Moscou
4(1937),
388-394.
4.
S.
HELGASON,
Differential
Geometry
and
Symmetric
Spaces,
Academic
Press,
New
York,
1962.
5.
S.
KOBAYASHI
AND
K.
NOMIZU,
Foundations
of
Differential
Geometry,
vol.
II,
Interscience
Publ.,
New
York,
1969.
6.
n.
F.
LAX,
The
cohomology
ring
of
SO(2n
/
2)/SO(2)XSO(2n)
and
some
geometrical
applica-
tions,
Proc.
Symp.
Pure
Math.,
Amer.
Math.
Soc.
27(1975),
361-362.
7.
T.
NAGANO,
Transformation
groups
with
(n
1)-dimensional
orbits
on
non-compact
mani-
folds,
Nagoya
Math.
J.
14(1959),
25-38.
8.
A.
WEIL,
Introduction
d
l’dtude
des
varidts
kiihlriennes,
Hermann,
Paris,
1958.
CHEN;
DEPARTMENT
OF
MATHEMATICS,
MICHIGAN
STATE
UNIVERSITY,
EAST
LANSING,
MICH-
IGAN
48824
NAGANO;
DEPARTMENT
OF
MATHEMATICS,
UNIVERSITY
OF
NOTRE
DAME,
NOTRE
DAME,
IN-
DIANA
46556