Graphics Reference
In-Depth Information
8.12.9. Lemma.
Let w be an n-form on a Riemannian n-manifold
M
n
. Let
p
Œ
M
.
If
v
1
,
v
2
,...,
v
n
and
w
1
,
w
2
,...,
w
n
are orthonormal bases of T
p
(
M
), then
( (
)
±
( (
)
w
pv v
,
,...,
v
=
w
pw w
,
,...,
w
.
12
n
1
2
n
Proof.
This follows from Lemma 8.12.8 and the fact that the matrix (a
ij
) in equa-
tion (8.37) that relates the bases is an orthogonal matrix whenever the bases are ortho-
normal bases, so that its determinant is ±1.
8.12.10. Lemma.
Let (
M
n
,s) be an oriented Riemannian n-manifold. For each
p
Œ
M
, choose an orthonormal basis
v
1,
p
,
v
2,
p
,...,
v
n,
p
of T
p
(
M
), so that s(
p
) = [
v
1,
p
,
v
2,
p
,
...,
v
n,
p
]. There is a unique (nonzero) n-form w on
M
with the property that
( (
)
=
w
pv
,
v
,...,
v
1
.
(8.39)
1
,
pp
2
,
n
,
p
Proof.
By Lemma 8.12.8, for any n-form w, the value w(
p
)(
v
1
,
v
2
,...,
v
n
) is the same
for all orthonormal bases
v
1
,
v
2
,...,
v
n
of T
p
(
M
) with s(
p
) = [
v
1
,
v
2
,...,
v
n
]. Further-
more, if a
1
, a
2
,..., a
n
is the dual basis for one such basis
v
1
,
v
2
,...,
v
n
, then a
i
(
v
j
) =
d
ij
and Property (4.25d) of the wedge product implies that (a
1
Ÿa
2
Ÿ ...Ÿa
n
)(
v
1
,
v
2
,
...,
v
n
) = 1. It follows easily that we can define w by the condition that w(
p
) =a
1,
p
Ÿa
2,
p
Ÿ
...Ÿa
n,
p
, where the a
i,
p
are a dual basis for the
v
i,
p
. If (
U
,j) is a coordinate neigh-
borhood for
M
, then w will have the form c du
1
Ÿdu
2
Ÿ ...Ÿdu
n
over this neighborhood
for some constant c. The uniqueness of the n-form w follows easily from the fact that
L
n
(T
p
(
M
)) ª
R
.
Definition.
Given an oriented Riemannian manifold
M
n
, the unique n-form w
defined by equation (8.39) in Lemma 8.12.10 is called the
volume element
of
M
and
is usually denoted by dV, except in the case of surfaces where one typically writes dA
and in the case of curves where one writes ds.
We shall see shortly why the volume element has the name it has.
Lemma 8.12.8 implies that an n-form w on
R
n
divides the set of ordered bases of
R
n
into two subsets in the same way that we defined an orientation of
R
n
, namely,
one set consists of those bases on which it is greater than zero and the other one con-
sists of those on which it is less than zero. Lemma 8.12.10 shows how an orientation
defines a nonzero form. This leads to the following observation that gives us another
way of thinking about orientation:
An n-manifold
M
n
8.12.11. Theorem.
is orientable if and only if it has a nonzero
n-form.
Proof.
Assume that
M
has a nonzero n-form w. If
p
Œ
M
, then choose the orienta-
tion of T
p
(
M
) on whose representatives w is greater than zero. We leave it to the reader
to show that this defines a continuously varying choice of orientations of the tangent
planes. Conversely, assume that
M
is orientable. Endow
M
with a Riemannian metric,
choose an orientation s, and apply Lemma 8.12.10.
Moving on to the question of how to piece together local definitions of an inte-
gral on a manifold in a consistent way, the next lemma gives us exactly what we need.
