392 Submanifolds and Holonomy
case, the surface is called regular). As in the case of curves, we will be considering
smooth vector fields X along f .ThenwehaveX(s,t) ∈ T
f (s,t)
M for all (s,t) ∈ U .
As usual, we will denote by
D
∂
s
the covariant derivative along the curve s → f (s,t)
with t fixed. The corresponding tangent vector field of this curve is denoted by
∂
f
∂
s
.
In the same way, we define
D
∂
t
and
∂
f
∂
t
. From the fact that th e Levi-Civita covariant
derivative is torsion-free, we deduce
D
∂
s
∂
f
∂
t
=
D
∂
t
∂
f
∂
s
.
However, the covariant derivatives with respect to s and t do not commute in general
if the curvature tensor does not vanish. More precisely,
D
∂
s
D
∂
t
X −
D
∂
t
D
∂
s
X = R
∂
f
∂
s
,
∂
f
∂
t
.
We will often omit f in the partial derivatives and simply write
∂
∂
s
and
∂
∂
t
.
Note that, when a smooth curve c(t) is defined on a closed interval [a, b],this
means that c(t) is the restriction to [a , b] of a smooth curve that is defined on an open
interval I with [a,b] ⊂ I. A similar rem ark applies to a surface that is defined on a
closed subset of R
2
.
Geodesics
Of great importance in Riemannian geometry are d istance-minimizing curves.
Distance-minimizing curves between two points in a Riemannian manifold may not
exist. However, they do exist provided the manifold is connected and complete.
Distance-minimizing curves
γ
are solutions of a variational problem. The corre-
sponding first variation formula shows that any such curve
γ
satisfies
˙
γ
=
D
dt
˙
γ
= 0.
A smooth curve
γ
satisfying this eq uation is called a geodesic. Every geodesic is
locally distance-minimizing, but not globally, as a g reat circle on a sphere illustrates.
The basic theory of ordinary differential equations implies that for every point
p ∈ M and every tangent vector v ∈ T
p
M there exists a unique geodesic
γ
: I → M
with 0 ∈ I,
γ
(0)=p,
˙
γ
(0)=v, and such that for any other geodesic
α
: J → M with
0 ∈ J,
α
(0)=p and
˙
α
(0)=v we have J ⊂ I. This curve
γ
is often called the maximal
geodesic in M through p tangent to v. We denote this maximal geodesic sometimes
by
γ
v
and its maximal domain by I
v
.
The Hopf-Rinow Theorem states that a Riemannian manifold is complete if and
only if I
v
= R for all v ∈TM.
Exponential map and normal coordinates
The exponential map exp of M is the map
exp :
*
TM = {v ∈ TM :1∈ I
v
}→M , v →
γ
v
(1).
Basic Material 393
For p ∈ M the exponential map exp
p
is the map
exp
p
:
*
T
p
M = {v ∈ T
p
M :1∈ I
v
}→M , v →
γ
v
(1).
There exists an open neighborhood U of 0 ∈ T
p
M such that the restriction of exp
p
to
U is a diffeomorphism into M. If we choose an orthonomal basis e
1
,...,e
m
of T
p
M,
then the map
(x
1
,...,x
m
) → exp
p
m
∑
i=1
x
i
e
i
defines local coordinates of M in some open neighborhood of p. Such coordinates
are called normal coordinates.
Jacobi fields
Let
γ
: I → M be a geodesic. A vector field Y along
γ
is called a Jacobi field if it
satisfies the linear second order ordinary differential equation
Y
+ R(Y,
˙
γ
)
˙
γ
= 0.
Standard theory of ordinary differential equations implies that the Jacobi fields along
a geodesic form a 2m-dimensional real vector space, where m = dimM. Every Jacobi
field is uniquely determined by the initial values Y (t
0
) and Y
(t
0
) at some t
0
∈ I.
Jacobi fields arise geometrically as infinitesimal variational vector fields of geodesic
variations.
Jacobi fields can be used to compute the differential of the exponential map.
Indeed, let p ∈ M and exp
p
be the exponential map of M at p.Forv ∈ T
p
M we
identify T
v
(T
p
M) with T
p
M in the canonical way. Then, for each w ∈ T
p
M,wehave
d
v
exp
p
(w)=Y
w
(1),
where Y
w
is the Jacobi field along
γ
v
with initial values Y
w
(0)=0andY
w
(0)=w.
Sectional curvature
A geometric inter pretation of the Riemannian curvature tensor can be given u sing
the sectional curvature. Consider a 2-dimensional linear subspace V of T
p
M, p ∈ M,
and choose an orthonormal basis X ,Y of V . Since the exponential map exp
p
: T
p
M →
M is a local diffeomorphism near 0 in T
p
M, it maps an open neighborhood of 0 in V
onto a 2-dimensional surface S in M. Then the Gaussian curvature of S at p,which
we denote by K(V ), satisfies
K(V )=R(X,Y )Y,X.
Let G
2
(TM) be the Grassmann bundle over M consisting of all 2-dimensional linear
subspaces V ⊂ T
p
M, p ∈ M.Themap
K : G
2
(TM) → R , V → K(V )
394 Submanifolds and Holonomy
is called sectional curvature function of M,andK(V ) is called the sectional curva-
ture of M with respect to
σ
. It is worthwhile to mention that one can reconstruct
the Riemannian curvature tensor from the sectional curvature function by using the
curvature identities.
A Riemannian manifold M is said to have constant curvature if the sectional cur-
vature function is constant. If M is connected and dimM ≥ 3, the second Bianchi
identity and Schur’s Lemma imply the following well-known result: If the sectional
curvature function depends only on the point p, then M has constant curvature.A
space of constant curvature is also called a space form or real space form.TheRie-
mannian curvature tensor of a space form with constant curvature
κ
∈ R is given
by
R(X,Y )Z =
κ
(Y,ZX −X, ZY ).
Every connected 3-dimensional Einstein manifold is a space form. It is an algebraic
fact (that is, does not involve the second Bianchi identity) that a Riem annian manifold
M has constant sectional curvature equal to zero if and only if M is flat, that is, the
Riemannian curvature tensor of M vanishes.
A connected, simply connected, complete Riemannian manifold with nonpos-
itive sectional curvature is called a Hadamard manifold .TheHadamard Theo-
rem states that for each point p in a Hadamard manifold M the exponential map
exp
p
: T
p
M → M is a diffeomorphism. More generally, if M is a connected, complete
Riemannian manifold with nonpositive sectional curvature, then the exponential map
exp
p
: T
p
M → M is a covering map for each p ∈ M.
Distributions and the Frobenius Theorem
A distribution on a Riemannian manifold M is a smooth vector subbundle H of
the tangent bundle TM. A distribution H on M is called integrable if for each p ∈M
there exists a connected submanifold L
p
of M such that T
q
L
p
= H
q
for all q ∈ L
p
.
Such a submanifold L
p
is called an integral manifold of H .TheFrobenius Theorem
states that H is integrable if and only if it is involutive, that is, if the Lie bracket
of any two vector fields tangent to H is again a vector field tangent to H .IfH is
integrable, there exists through each point p ∈ M a maximal integral manifold of H
containing p. Such a maximal integral manifold is called the leaf of H through p.
A distribution H on M is called autoparallel if ∇
H
H ⊂ H ,thatis,ifforany
two vector fields X ,Y tangent to H the vector field ∇
X
Y is also tangent to H .Bythe
Frobenius Theorem every autoparallel distribution is integrable. An integrable distri-
bution on M is autoparallel if and only if its leaves are totally geodesic submanifolds
of M. A distribution H on M is called parallel if ∇
X
H ⊂ H for all vector fields
X on M. Obviously, every parallel distribution is autoparallel. Since ∇ is a metr ic
connection, for each parallel distribution H on M, its orthogonal complement H
⊥
in TM is also a parallel distribution on M.
Holonomy
A Riemannian manifold M is said to be flat if its Riemannian curvature tensor
vanishes. This implies that, locally, the parallel transport does not depend on the
Basic Material 395
curve used for joining two given points. If the Riemannian curvature tensor does
not vanish, the parallel transport depends on the curve. A measure for a Riemannian
manifold for the deviation from being flat is given by the holonomy group.
Let p ∈ M and Ω(p) be the set of all piecewise smooth curves c : [0,1] →M with
c(0)=c(1)=p. Then the parallel translation along any curve c ∈ Ω(p) from c(0) to
c(1) is an orthogonal transformation of T
p
M. In an obvious manner, all these parallel
translations generate a subgroup Hol
p
(M) of the orthogonal group O(T
p
M),which
is called the holonomy group of M at p. As a subset of O(T
p
M), it carries a natural
topology. With respect to this topology, the identity component Hol
o
p
(M) of Hol
p
(M)
is called the restricted holonomy g roup of M at p. The restricted holonomy group is
generated by all those transformations arising from null homotopic curves in Ω(p).
If M is connected, then all (restricted) holonomy groups are congruent to each
other, and, in this situation, one speaks of the (restricted) holonomy group of the
manifold M, which we will then denote by Hol(M) resp. Hol
o
(M). The connected Lie
group Hol
o
(M) is always compact, whereas Hol(M) is, in general, not closed in the
orthogonal group. A reduction of the holonomy group corresponds to an additional
geometric structure on M. For example, Hol(M) is contained in SO(T
p
M) for some
p ∈ M if and only if M is orientable. An excellent introduction to holonomy groups
can be found in the book by Salamon [289].
K
¨
ahler manifolds
An almost complex structure on a smooth manifold M is a tensor field J of type
(1,1) on M satisfying J
2
= −id
TM
.Analmost complex manifold is a smooth manifold
equipped with an almost complex structure. Each tangent space of an almost complex
manifold is isomorphic to a complex vector space, which implies that the dimension
of an almost complex manifold is an even number.
A Hermitian metric on an almost complex manifold M is a Riemannian metric
·,· for which the almost complex structure J on M is orthogonal, that is,
JX ,JY= X,Y
for all X ,Y ∈T
p
M, p ∈M. An orthogonal almost complex structure on a Riemannian
manifold is called an almost Hermitian structure.
Every complex manifold M has a canonical almost complex structure. In fact, if
z = x + iy is a local coordinate on M,define
J
∂
∂
x
ν
=
∂
∂
y
ν
, J
∂
∂
y
ν
= −
∂
∂
x
ν
.
These local almost complex structures are compatible on the intersection of any two
coordinate neighborhoods and hence induce an almost complex structure, which
is called the induced complex structure on M. An almost complex structure J on
a smooth manifold M is integrable if M can be equipped with the structure of a
complex manifold so that J is the induced complex structure. A famous result by
Newlander-Nirenberg says that the almost complex structure J of an almost complex
manifold M is integrable if and only if
[X,Y ]+J[JX ,Y ]+J[X, JY] −[JX , JY ]=0
396 Submanifolds and Holonomy
for all X,Y ∈ T
p
M, p ∈ M.AHermitian manifold is an almost Hermitian manifold
with an integrable almost complex structure. The almost Hermitian structure of a
Hermitian man ifold is called a Hermitian structure.
The 2-form
ω
on a Herm itian manifold M defined by
ω
(X,Y )=X,JY
is called the K
¨
ahler form of M.AK
¨
ahler manifold is a Hermitian manifold whose
K¨ahler form is closed. In this situation the Hermitian structure is called a K
¨
ahler
structure. A Hermitian manifold M is a K¨ahler manifold if and o nly if its Hermitian
structure J is parallel with respect to the Levi-Civita covariant d erivative ∇ of M,
that is, if ∇J = 0. The latter condition characterizes the K¨ahler manifolds among all
Hermitian man ifolds by the geometric property that parallel translation along curves
commutes with the Hermitian structure J.
A2m-dimensional connected Riemannian manifold M can be equipped with the
structure of a K¨ahler manifold if and only if its holonomy group Hol(M) is contained
in the unitary group U
m
. Some standard examples of K¨ahler manifolds are the com-
plex vector space C
m
, the complex projective space CP
m
= SU
m+1
/S(U
1
U
m
) (m ≥2)
and the complex hyperbolic space CH
m
= SU
1,m
/S(U
1
U
m
) (m ≥ 2).
The Riemannian curvature tensor R and the Ricci tensor Ric of a K¨ahler m anifold
M satisfy
R(X,Y )JZ = JR(X ,Y)Z and Ric(JX )=JRic(X)
for all X,Y, Z ∈ T
p
M, p ∈ M.
Quaternionic K
¨
ahler manifolds
A quaternionic K
¨
ahler structure on a Riemannian manifold M is a rank three
vector subbundle J of the endomorphism bundle End(TM) over M with the following
properties:
(1) For each p in M there exist an open neighborhood U of p in M and sections
J
1
,J
2
,J
3
of J over U so that J
ν
is an almost Hermitian structure on U and
J
ν
J
ν
+1
= J
ν
+2
= −J
ν
+1
J
ν
(index modulo three)
for all
ν
= 1, 2, 3;
(2) J is a parallel subbundle of End(TM),thatis,ifJ is a section in J and X a
vector field on M,then∇
X
J is also a section in J.
Each triple J
1
,J
2
,J
3
of the above kind is called a canonical local basis of J,or,ifre-
stricted to the tangent space T
p
M of M at p,acanonical basis of J
p
.Aquaternionic
K
¨
ahler manifold is a Riemannian manifold equipped with a quaternionic K¨ahler
structure. The canonical bases of a quaternionic K¨ahler structure turn the tangent
spaces of a quaternionic K¨ahler manifold into quaternionic vector spaces. Therefore,
the dimension of a quaternionic K¨ahler manifold is 4m for some m ∈ N.
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