6 Submanifolds and Holonomy
Codazzi equation:
(∇
⊥
X
α
)(Y, Z)=(∇
⊥
Y
α
)(X,Z);
Ricci equation:
R
⊥
(X,Y )
ξ
,
η
= [A
ξ
,A
η
]X,Y .
The fundamental equations of Gauss, Codazzi, and Ricci play an analogous role
in submanifold geometry of space fo rms as the Frenet equations in the differential
geometry of curves. Namely, they suffice to determine, up to isometries o f the am bi-
ent space, a submanifold of a space form. This is the conclusion of the Fundamental
Theorem of Local Submanifold Geometry.
Theorem 1.1.2 (Fundamental Theorem of Local Submanifold Geometry) Let M
be an m-dimensional Riemannian manifold,
ν
a Riemannian vector bundle over M
of rank r, ∇
a metric connection on
ν
, and
α
(X,Y ) a symmetric tensor field on M
with values in
ν
.Define A :
ν
→ End(TM) by
A
ξ
X,Y =
α
(X,Y ),
ξ
for X ,Y ∈ T
p
M,
ξ
∈
ν
p
, p ∈ M.
Assume that
α
, A, and ∇
satisfy the equations of Gauss, Codazzi, and Ricci for some
κ
∈ R. Then, for each point p ∈ M, there exists an open neighborhood U of p in M
and an isometric immersion
f : U →
¯
M
n
(
κ
)
from U into a space form
¯
M
n
(
κ
) with constant curvature
κ
,n= m + r , such that
α
and A are the second fundamental form and shape operator of f , respectively, and
ν
is isomorphic to the normal bundle of f . The immersion f is unique up to an isometry
of
¯
M
n
(
κ
). Moreover, if two isometric immersions have the same second fundamental
form and normal connection, then they locally coincide up to an isometry of the
ambient space.
Proof We give a proof for
κ
= 0, that is, for
¯
M
n
(
κ
)=R
n
. The proof for the general
case is similar and can be found, for instance, in [299].
Let E = TM⊕
ν
be the Whitney sum of the Riemannian vector bundles TM and
ν
over M.Wedefine a connection
ˆ
∇ on E by
ˆ
∇
X
Y = ∇
X
Y +
α
(X,Y ) and
ˆ
∇
X
ξ
= −A
ξ
X + ∇
X
ξ
for all vector fields X ,Y on M and sections
ξ
in
ν
. Then the Gauss-Codazzi-Ricci
equations imply that
ˆ
∇ is a flat connection, that is, the curvature tensor of
ˆ
∇ vanishes.
Thus, there exists an open neighborhood V of p in M and a
ˆ
∇-parallel frame field
ξ
1
,...,
ξ
n
of E over V . Such a frame field is unique up to a linear isometry of R
n
.We
denote by
η
i
the 1-form on V defined by
η
i
(X)=
ξ
i
,X.Thenwehave
d
η
i
(X,Y )=X
ξ
i
,Y −Y
ξ
i
,X−
ξ
i
,[X,Y ]
=
ˆ
∇
X
ξ
i
,Y +
ξ
i
,∇
X
Y +
ξ
i
,
α
(X,Y )
−
ˆ
∇
Y
ξ
i
,X−
ξ
i
,∇
Y
X−
ξ
i
,
α
(Y,X)−
ξ
i
,[X,Y ] = 0,
Basics of Submanifold Theory in Space Forms 7
since by construction the TM-part of
ˆ
∇ coincides with the Levi-Civita connection on
M, and because
α
is symmetric by assumption. Thus,
η
i
is closed and hence there
exists a smooth function f
i
: U
i
→ R on some open neighborhood U
i
of p in M such
that df
i
=
η
i
.LetU = U
1
∩...∩U
n
.Then f =(f
1
,..., f
n
) : U → R
n
is an isometric
immersion with the required properties. Note that once we fix the frame field, f is
unique up to translation, so f is unique up to an isometry of R
n
. Note that there is
a bundle isomorphism between E and T R
n
restricted to U, which is the identity on
TM.
Finally, assume that two isometric immersions have the same second fundamental
form and normal connection. We define as above Riemannian vector bundles E and
E
over M. Then there are bundle isomorphisms between E and T R
n
|
M
and E
and
T R
n
|
M
, wh ich are the identity on TM and preserve both metric and connection. Using
the same arguments as in the first part of the proof we can see that the immersions
differ locally by a rigid motion of R
n
.
1.1.3 Equations of higher order
The fundamental equations of first and second order are the basic tools for in-
vestigating the geometry of submanifolds. However, one can derive further useful
equations of higher order. We discuss here an example of a third order equation.
To begin with, recall that the second covariant derivative o f the second fundamental
form is given by
(∇
2
X
1
X
2
α
)(X
3
,X
4
)=∇
⊥
X
1
(∇
⊥
X
2
α
)(X
3
,X
4
) −(∇
⊥
∇
X
1
X
2
α
)(X
3
,X
4
)
−(∇
⊥
X
2
α
)(∇
X
1
X
3
,X
4
) −(∇
⊥
X
2
α
)(X
3
,∇
X
1
X
4
).
Then, taking the covariant derivative ∇
⊥
X
1
of the equation
(∇
⊥
X
2
α
)(X
3
,X
4
)=∇
⊥
X
2
α
(X
3
,X
4
) −
α
(∇
X
2
X
3
,X
4
) −
α
(X
3
,∇
X
2
X
4
),
a straightforward computation yields the so-called Ricci formula
∇
2
X
1
X
2
α
−∇
2
X
2
X
1
α
= −
ˆ
R(X
1
,X
2
) ·
α
.
The curvature operator
ˆ
R(X
1
,X
2
) acts on the tangent space as the Riemannian cur-
vature tensor and on the normal space as the normal curvature tensor. The notation
ˆ
R(X
1
,X
2
) ·
α
means that
ˆ
R(X
1
,X
2
) acts on the
α
as a derivation.
Remark 1.1.3 By taking the trace of the operator ∇
2
one defines the Laplace-
Beltrami operator Δ. For example,
Δ
α
= tr(∇
2
α
).
Below we will use some formulae involving Δ
α
, for instance (cf. [83, formula
(3.12)]):
1
2
Δ
α
2
= ∇
⊥
α
2
+
α
,Δ
α
,
8 Submanifolds and Holonomy
where the norms and inner product are the usual ones for tensors. The term
α
,Δ
α
can be computed directly in terms of the second fundamental form and the normal
curvature tensor. This was done, for instance, in the paper by Chern, do Carmo,
and Kobayashi [83] (as mentioned in the introduction) for the case of a minimal
submanifold of a space form of constant curvature
κ
. In this case, the relation is
α
,Δ
α
= n
κ
α
2
−
α
◦
α
t
2
−R
⊥
2
,
where
α
t
is the adjoint of
α
regarded as a homomorphism from TM⊕TM to
ν
M.
1.2 Models of space forms
A large part of this book deals with problems in space forms. For this reason, we
now take a closer look at the standard models of these spaces. The application of the
fundamental equations simplifies their description.
1.2.1 The Euclidean s pace R
n
Consider R
n
as an n-dimensional smooth manifold equipped with the standard
smooth structure. At each point p ∈ R
n
we identify the tangent space T
p
R
n
of R
n
at
p with R
n
by means of the isomorphism
T
p
R
n
→ R
n
,
˙
γ
v
(0) → v ,
where
γ
v
(t)=p +tv. Using this isomorphism, we get an inner product ·, · on T
p
R
n
by the usual dot product on R
n
,thatis,
v, w =
n
∑
i=1
v
i
w
i
.
This family of inner products defines a Riemannian metric ·,· on R
n
. We call R
n
equipped with this Riemannian metric the n-dimensional Euclidean space,whichwe
also denote by R
n
or sometimes by E
n
. By means of the above isomorphism, the
Levi-Civita connection ∇ of R
n
coincides with the usual derivative D of R
n
.Itis
then a straightforward exercise to check that the Riemannian curvature tensor of R
n
vanishes.
The isometry group I(R
n
) of R
n
is the semidirect product
I(R
n
)=O
n
R
n
,
where R
n
acts on itself by left translations. Explicitly, the action of I(R
n
) is given by
(O
n
R
n
) ×R
n
→ R
n
, ((A,a),x) → Ax + a
Basics of Submanifold Theory in Space Forms 9
and the group structure of I(R
n
) is given by the formula
(A,a) ·(B, b)=(AB,Ab + a).
The identity component I
o
(R
n
) of I(R
n
) is SO
n
R
n
and the quotient group
I(R
n
)/I
o
(R
n
) is isomorphic to Z
2
.
1.2.2 The sphere S
n
(r)
Let r be a positive real number and consider the sphere
S
n
(r)={p ∈ R
n+1
: p, p = r
2
}
with radius r and center 0 in R
n+1
. It is a smooth submanifold of R
n+1
with a unit
normal vector field
ξ
defined by
ξ
p
=
1
r
p,
where we again use the canonical isomorphism T
p
R
n
∼
=
R
n
. Differentiating
ξ
with
respect to tangent vectors of S
n
(r), we obtain for the shape operator A
ξ
of S
n
(r) with
respect to
ξ
the expression
A
ξ
X = −
1
r
X
for each tangent vector X of S
n
(r). The Gauss equation then gives us the Riemannian
curvature tensor R of S
n
(r), namely
R(X,Y )Z =
1
r
2
(Y,ZX −X, ZY ).
This implies that S
n
(r) has constant sectional curvature r
−2
. We usually denote the
unit sphere S
n
(1) by S
n
.
The isometry group I(S
n
(r)) of S
n
(r) is the orthogonal group O
n+1
acting on
S
n
(r) in the obvious way. The identity component I
o
(S
n
(r)) of I(S
n
(r)) is SO
n+1
,
and the quotient group I(S
n
(r))/I
o
(S
n
(r)) is isomorphic to Z
2
.
1.2.3 The hyperbolic space H
n
(r)
There are various models for the hyperbolic space. One of them is constructed in
a similar way to the sphere, but starting from a Lorentzian space. We will refer to it
as the standard model. Consider R
n+1
equipped with the bilinear form
v, w =
n
∑
i=1
v
i
w
i
−v
n+1
w
n+1
of signature (n,1). Identifying each tangent space of R
n+1
with R
n+1
as described
above, we get a Lorentzian metric on R
n+1
, which we also denote by ·,·.The
10 Submanifolds and Holonomy
smooth manifold R
n+1
equipped with this Lorentzian metric is called Lorentzian
space and will be denoted by R
n,1
.Letr be a positive real number and
H
n
(r)={p ∈ R
n,1
: p, p = −r
2
, p
n+1
> 0}.
This is a connected smooth submanifold of R
n,1
with timelike unit normal vector
field
ξ
p
=
1
r
p.
The tangent space T
p
H
n
(r) consists of all vectors orthogonal to
ξ
p
and h ence is
a spacelike linear subspace of R
n,1
. Thus, the Lorentzian metric of R
n,1
induces a
Riemannian metric on H
n
(r).
An affine subspace W of R
n,1
is Riemannian, Lorentzian, or degenerate if the
restriction of the inner product ·,·on R
n,1
to the vector part of W is positive definite,
has signature (dim(W ) −1,1), or is d egenerate, respectively.
The shape operator of H
n
(r) with respect to
ξ
is given by
A
ξ
X = −
1
r
X
for all tangent vectors X of H
n
(r). The Gauss equation, which is valid also in the
Lorentzian situation , then gives f or the Riemannian curvature tensor R of H
n
(r) the
expression
R(X,Y )Z = −
1
r
2
(Y,ZX −X, ZY ).
It follows that H
n
(r) has constant sectional curvature −r
−2
. We write H
n
instead of
H
n
(1).
The orthogonal group O
n,1
of all transformations of R
n,1
preserving the
Lorentzian inner product consists of four connected components, depending on
whether the determinant is 1 or −1 and the transformation is time-preserving or
time-reversing. The time-preserving transformations in O
n,1
are those that leave
H
n
(r) invariant and form the isometry group I(H
n
(r)) of H
n
(r). The identity com-
ponent I
o
(H
n
(r)) is SO
o
n,1
and the quotient group I(H
n
(r))/I
o
(H
n
(r)) is isomorphic
to Z
2
×Z
2
.
Several other classical models of hyperbolic space are very useful for visualizing
geometric aspects of H
n
, for instance, for visualizing geodesics. We briefly mention
two of them.
The first of these models is known as the half space model, which consists of the
the half space
{x =(x
1
,...,x
n
) ∈ R
n
: x
n
> 0}
endowed with the Riemannian metric
ds
2
=
1
x
2
n
(dx
2
1
+ ...+ dx
2
n
).
In this model, the geodesics are either lines orthogonal to the hyperplane x
n
= 0or
circles intersecting the hyperplane x
n
= 0 orthogonally.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.