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If the surface
S
is oriented, then the integral of this volume element will give the usual
area of the surface. On the other hand, this volume element is also defined for the
Moebius strip, which we know is a nonorientable surface. In this case one can show
that the volume element is
not
a differential form, since a nonorientable n-manifold
cannot have a nonzero n-form.
The generalization of the volume element defined by equation (8.44) to n-
manifolds is discussed in [Spiv70a]. See also Section 9.8.
Finally, there is a direct connection between differential forms and cohomology.
Let
M
n
be a differentiable manifold.
Definition.
A differential form w on M is called
closed
if dw=0. It is called
exact
if
w=dh for some differential form h.
If we denote the vector space of closed k-forms on
M
by Z
k
(d)
(
M
) and the vector
space of exact forms by B
k
(d)
(
M
), then B
k
(d)
(
M
) is a subspace of Z
k
(d)
(
M
) because
d
2
= 0. It follows that we can define
k
()
()
Z
B
M
M
(
d
)
k
()
=
H
M
.
(
)
d
k
(
)
d
Definition.
The vector space H
k
(d)
(
M
) is called the
kth de Rham cohomology group
or vector space
of
M
.
A very important result is the fact that the de Rham cohomology groups of a dif-
ferentiable manifold agree with the ordinary cohomology groups with real coefficients
(Section 7.3 described cohomology with integer coefficients), that is,
()
()
ª
k
k
(
)
H
M
H
;
.
d
This shows that the differential forms of a manifold are intimately tied to the topol-
ogy of the space. Unfortunately, there is no space to expand on that connection here.
See [Spiv70a] or [GuiP74].
The Manifold P
n
8.13
Again, because projective space is such an interesting and important space, we devote
a separate section to study its properties as a differentiable manifold.
First of all, we show that
P
n
is a differentiable manifold. We need to show all of
its points have coordinate neighborhoods. For i = 1, 2,..., n + 1, define subsets
U
i
in
P
n
by
=
[
{
]
}
U
i
cc
,
,...,
c
c
π
0
(8.45)
12
n
+
1
i
and
