Submanifolds of Riemannian Manifolds 275
This shows that T
q
M is obtained by
¯
∇-parallel translation of V along
γ
v
from p to q.
It is now sufficient to p rove that
¯
∇-parallel translation along loops in M based at
p leaves V invariant. For the moment let us assume that this is true. Let q ∈ M and
v ∈V ∩U
ε
(0) so that q = exp
p
(v).Letc be any loop in M based at q. We get a loop ˆc
based at p by running along
γ
v
from p to q first, then along the loop c, and then back
to p along
γ
v
. As we already saw above, T
q
M is the
¯
∇-parallel translate of V along
γ
v
from p to q.Since
¯
∇-parallel translation along loops in M based at p preserves V ,it
follows that
¯
∇-parallel translation along c preserves T
q
M.Nowletc : [0,1] → M be
any curve in M. From each point c(t) we construct a loop by running first from c(0)
to c(t) along c, and then along the radial geodesic from c(t) to p,andfinally along
the rad ial geode sic from p to c(0). The invariance of the tangent spaces of M with
respect to
¯
∇-parallel transport along loops and along the radial geodesics implies that
T
c(t)
M is obtained by
¯
∇-parallel transport of T
c(0)
M along c. Hence we have shown
that
¯
∇-parallel transport along curves in M leaves the tangent spaces of M invariant.
This implies that the induced connection on M coincides with the restriction of
¯
∇ to
tangent vector fields of M. From the Gauss formula we finally get that the second
fundamental form of M vanishes, and hence M is totally geodesic in
¯
M.
Thus, it remains to prove that
¯
∇-parallel translation along loops in M based at p
leaves V invariant. We first prove the following:
Lemma 10.3.4 Let
¯
M be a Riemannian manifold and p ∈
¯
M. Let
f : [0,
δ
] ×[0,1] →
¯
M
be a smooth map with f (s,0)=p for all s ∈ [0 ,
δ
]. For all s ∈ [0,
δ
] and t ∈ [0,1] we
define
f
s
: [0,1] →
¯
M , t → f (s,t) and f
t
: [0,
δ
] →
¯
M , s → f (s,t).
For every s ∈ [0,
δ
] we denote by
τ
(s) ∈ SO(T
p
¯
M) the orthogonal transformation of
T
p
¯
M obtained by parallel translation along f
0
from p = f
0
(0) to f
0
(1)= f
1
(0),then
along f
1
from f
1
(0) to f
1
(s)= f
s
(1), and finally along f
s
from f
s
(1) to f
s
(0)=p.
Let A(s) ∈ so(T
p
¯
M) be the skewsymmetric transformation of T
p
¯
Mdefined by A(s)=
τ
(s) ◦
τ
(s)
−1
for all s ∈ [0,
δ
]. Then, for each u,w ∈ T
p
¯
M, we have
A(s)u,w =
1
0
¯
R
∂
f
∂
s
(s,t),
∂
f
∂
t
(s,t)
U
s
(t),W
s
(t)dt ,
where U
s
(t) and W
s
(t) are the parallel vector fields along f
s
with U
s
(0)=u and
W
s
(0)=w, respectively.
Proof Let s ∈ (0,
δ
).Wedefine the smooth map
˜
f : [0,
δ
−s] ×[0,1] →
¯
M , ( ˜s,t) → f (s + ˜s,t).
To avoid confusion, we denote the objects associated to
˜
f with tilded symbols˜. We
have
τ
(s + ˜s)=
˜
τ
( ˜s) ◦
τ
(s)
276 Submanifolds and Holonomy
and
A(s) ◦
τ
(s)=
τ
(s)=
˜
τ
(0) ◦
τ
(s)=
˜
A(0) ◦
τ
(s),
since
˜
τ
(0) is the identity transformation of T
p
¯
M. This implies A(s)=
˜
A(0).
Thisshowsthatitsuffices to prove the above formula only for s = 0, because
then
A(s)u,w=
˜
A(0)u,w =
1
0
¯
R
∂
˜
f
∂
˜s
(0,t),
∂
˜
f
∂
t
(0,t)
˜
U
0
(t),
˜
W
0
(t)dt
=
1
0
¯
R
∂
f
∂
s
(s,t),
∂
f
∂
t
(s,t)
U
s
(t),W
s
(t)dt .
For s =
δ
the formula follows by a continuity argument.
Let U(s,t) be the vector field along f (s,t) obtained by parallel translation of u
along f
0
from p = f
0
(0) to f
0
(1)= f
1
(0), then along f
1
from f
1
(0) to f
1
(s)= f
s
(1),
and finally along f
s
from f
s
(1) to f
s
(t).Thenwehave
U(s,0)=
τ
(s)u
and
A(0)u =
τ
(0)u = Z(0),
where Z is the vector field along f
0
defined by
Z(t)=
D
∂
s
U
(0,t),
where D is the covariant derivative along curves associated with the connection
¯
∇.
By construction, the vector field t →U (s,t) is pa rallel along f
s
, which implies
Z
(t)=
D
∂
t
D
∂
s
U
(0,t)=
¯
R
∂
f
∂
t
(0,t),
∂
f
∂
s
(0,t)
U
0
(t).
For the smooth function
g(t)=Z(t),W
0
(t)
we therefore get
g
(t)=Z
(t),W
0
(t) =
¯
R
∂
f
∂
t
(0,t),
∂
f
∂
s
(0,t)
U
0
(t),W
0
(t)
with initial condition
g(1)=Z(1),W
0
(1) = 0,
since Z(1)=0 by construction. Using Barrow’s rule we then obtain
A(0)u,w = g(0 )=g(1) −
1
0
¯
R
∂
f
∂
t
(0,t),
∂
f
∂
s
(0,t)
U
0
(t),W
0
(t)dt
=
1
0
¯
R
∂
f
∂
s
(0,t),
∂
f
∂
t
(0,t)
U
0
(t),W
0
(t)dt ,
Submanifolds of Riemannian Manifolds 277
by which the lemma is proved.
Proof of Theorem 10.3.3 (continued): We still have to prove that V is invariant by
¯
∇-parallel translation along loops in M based at p.Letc : [0,1] → M be a loop in M
with c(0)=c(1)=p. Then there exists a unique curve
ξ
: [0,1] →V ∩U
ε
(0) ⊂ T
p
M
so that c = exp
p
◦
ξ
.Wedefine
f : [0,1] ×[0, 1] → M , (s,t) → exp
p
(t
ξ
(s))
and use the same notations as in the previous lemma. By assumption, the Riemannian
curvature tensor
¯
R of
¯
M preserves the
¯
∇-parallel tran slate of V along the geodesics f
s
.
In the first p art of the proof we saw that T
f (s,t)
M is obtained b y
¯
∇-parallel translation
of V along the geodesic f
s
from p = f
s
(0) to f
s
(t)= f (s,t). Combining these facts
with the equation in the previous lemma we obtain
A(s)u,w= 0
for all s ∈ [0,1], u ∈V and w ∈V
⊥
,thatis,
A(s) ∈ so(V ) ⊕so(V
⊥
)
for all s ∈ [0,1].Since
τ
(0) is the identity transformation of T
p
¯
M, we conclude that
τ
(s) ∈ O(V ) ×O (V
⊥
)
for all s ∈[0,1]. In particular,
τ
(1) ∈O(V )×O(V
⊥
).But
τ
(1) is, by construction, the
¯
∇-parallel translation along the loop c from c(0) to c(1). This concludes the proof of
Theorem 10.3.3.
If the manifold
¯
M is real analytic, the assumption on the geodesics can be re-
placed by the local property that the Riemannian curvature tensor
¯
R
p
and all its co-
variant derivatives (
¯
∇
k
¯
R)
p
, k ≥ 1, at p preserve V .
A global version of the existence of complete totally geodesic immersed sub-
manifolds of complete Riemannian manifolds was obtained by Hermann [152] us-
ing once-broken geodesics. Let p ∈
¯
M and V be a linear subspace of T
p
¯
M.Let
γ
: [0, b] →
¯
M be a once-broken geodesic starting at p and broken at t
o
∈ (0,b).
Following Hermann, we say that
γ
is V -admissible if
˙
γ
(t) lies in the parallel translate
of V along
γ
from p to
γ
(t) for all t ∈ [0,b] and if
γ
([t
o
,b]) is contained in some con-
vex neighborhood of
γ
(t
o
). It is convenient to encompass smooth geodesics among
once-broken geodesics.
Theorem 10.3.5 (Hermann) Let
¯
M be a complete Riemannian manifold, p ∈
¯
M and
V be a linear subspace of T
p
¯
M. Then there exists an immersed complete totally
geodesic submanifold M of
¯
M with p ∈ M and T
p
M = V if and only if for each
V -admissible once-broken geodesic
γ
: [0, b] →
¯
M the Riemannian curvature tensor
of
¯
Mat
γ
(b) preserves the parallel translate of V along
γ
from p to
γ
(b).
278 Submanifolds and Holonomy
Proof The “only if” part of the theorem is trivial. Let us assume that for each V -
admissible once-broken geodesic
γ
: [0, b] →
¯
M the Riemannian curvature tensor of
¯
M at
γ
(b) preserves the parallel translate of V along
γ
from p to
γ
(b). According to
Cartan’s Local Existence Theorem 10.3.3 there exists a connected totally geodesic
submanifold M of
¯
M with p ∈ M and T
p
M = V . Without loss of generality, we can
assume that M is maximal. The only point we have to keep in mind here is that M may
not be embedded but only immersed in
¯
M. We now assume that M is not complete
and derive a contradiction.
If M is not complete, there exists a geodesic
β
: [0, 1) →M for which lim
t→1
β
(t)
does not exist in M.Since
¯
M is complete and M is totally geodesic in
¯
M there exists
a geodesic
α
: [0,1] →
¯
M such that
α
(t)=
β
(t) for all t ∈ [0, 1).LetW be the
¯
∇-
parallel tra nslate of V along
α
from p to q =
α
(1). It follows from the assumption and
Theorem 10.3.3 that there exists a connected totally geodesic submanifold N of
¯
M
with q ∈ N and T
q
N = W . Now consider the once-broken geodesic
γ
: [0,1 +
ε
] →
¯
M
given by
γ
(t)=
β
(t)=
α
(t) for t ∈ [0,1),
γ
(1)=q =
α
(1) and
γ
(t)=
β
(1 −t)=
α
(1 −t) for t ∈ (1, 1 +
ε
),
ε
∈R
+
sufficiently small. By construction, the
¯
∇-parallel
translate of V along
β
from p to
β
(1 −
δ
) coincides with the parallel translate of W
along
γ
from q to
γ
(1 +
δ
)=
β
(1 −
δ
) for all
δ
∈ (0,
ε
). It follows that the tangent
spaces of M and N coincide at all points on
γ
((1,1 +
ε
)) =
β
((1 −
ε
,1)).Sincewe
assumed that M is maximal totally geodesic, rigidity of totally geodesic submanifolds
implies that N is contained in M. But this is a contradiction since q is in N but not in
M. It follows that M is complete.
We will discuss the existence problem for totally geodesic submanifolds of sym-
metric spaces in Section 11.1.
10.3.3 Fixed point sets of isometries
An important class of totally geodesic submanifolds is given b y fixed point sets
of isometries.
Proposition 10.3.6 Let f :
¯
M →
¯
M be an isometry of a Riemannian manifold
¯
M and
¯
M
f
= {p ∈
¯
M : f (p)=p}
be the set of fixed points of f . If
¯
M
f
= /0, then each connected component M of
¯
M
f
is
a totally geodesic embedded submanifold of
¯
M and for each p ∈ M we have
T
p
M = {X ∈ T
p
¯
M : d
p
f (X)=X}.
Proof Let M be a connected component of
¯
M
f
, p ∈ M and
V
p
= {X ∈ T
p
¯
M : d
p
f (X)=X}.
Since isometries map geodesics to geodesics, we have f ◦
γ
X
=
γ
X
for all X ∈ V
p
,
where
γ
X
is the maximal geodesic in
¯
M with
γ
X
(0)=p and
˙
γ
X
(0)=X. This implies
exp
p
(V
p
) ⊂ M,whereexp
p
: T
p
¯
M →
¯
M is the exponential map of
¯
M at p.Ifq ∈ M is
Submanifolds of Riemannian Manifolds 279
sufficiently close to p, say in some convex open neighborhood of p in
¯
M, there exists
a unique geodesic in this neighborhood connecting them. As p and q are fixed by f
and the geodesic between them is unique, the entire geodesic is fixed by f . Hence,
its tangent vector at p is fixed, and the geodesic is of the form
γ
X
with some X ∈V
p
.
Thus there exists an open neighborhood of p in M contained in exp
p
(V
p
).Asexp
p
is
a local diffeomorphism near 0 ∈ T
p
¯
M we can now conclude that M is an embedded
submanifold of
¯
M and T
p
M = V
p
(recall that p was a rbitrary). Next, let
γ
: I → M be a
geodesic in M with
γ
(0)=p and
˙
γ
(0)=X ∈T
p
M.AsT
p
M = V
p
and exp
p
(V
p
) ⊂ M,
uniqueness of geodesics implies that
γ
=
γ
X
|
I
,thatis,
γ
is a geodesic in
¯
M.
This proposition is of particular interest when f is an isometric involution on
¯
M.If
¯
M
f
is non-empty, then f is the geodesic reflection of
¯
M in each connected
component of
¯
M
f
. An interesting example is given by the geodesic symmetry s
p
of
a Riemannian symmetric space
¯
M at a given point p ∈
¯
M. The point p is an isolated
fixed point of s
p
. Each other connected component of the fixed point set of s
p
is
called a polar of
¯
M (with respect to p). Polars contain deep information about the
geometry and topology of a symmetric space; see, for example, [226, 227, 229–231].
10.3.4 The congruence problem for totally geodesic submanifolds
Another fundamental problem concerning totally geodesic submanifolds is con-
gruence. By this we mean the following: Given two Riemannian manifolds M and
¯
M
and two totally geodesic isometric immersions f
1
, f
2
: M →
¯
M, is there an isometry
g of
¯
M so that f
1
= g ◦ f
2
?Ifsuchag exists, the two immersions are said to be con-
gruent. A basic problem is to determine the congruence classes of totally geodesic
isometric immersions from a fixed Riemannian manifold M into another fixed Rie-
mannian manifold
¯
M. This is, in general, a rather difficult problem and has been
solved so far only for some particular ambient spaces
¯
M, for instance R
n
, Riemann-
ian symmetric spaces of rank one [345] or rank two [170–173].
10.4 Totally umbilical submanifolds and extrinsic spheres
Recall that a submanifold M of a Riemannian manifold
¯
M is said to be umbilical
in direction
ξ
if the shape operator A
ξ
of M with respect to the normal vector
ξ
is a multiple of the identity. If M is umbilical in any normal direction
ξ
,thenM is
called a totally umbilical submanifold of
¯
M.Themean curvature vector field H of a
submanifold M of
¯
M is defined by
H =
1
dim(M)
tr(
α
).
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