Graphics Reference
In-Depth Information
n
¥
[
Æ
n
h
:
S
01
,
S
by
(
)
=
(
)
+
(
)()
ht
p
,
cos
p
t
p
sin
p
t
s
p
.
If C(
p
),
p
Œ
S
n
, denotes the unit circle in the two-dimensional plane through the origin
with orthonormal basis consisting of the vectors
p
and s(
p
), then h(
p
,t) moves from
p
to -
p
on this circle C(
p
) as t moves from 0 to 1. Thus, h is a homotopy between the
identity map of
S
n
and the antipodal map. But the identity map has degree 1 and the
antipodal map has degree (-1)
2k+1
=-1 by Theorem 7.5.1.3. This contradiction shows
that s cannot exist.
8.5.6. Corollary.
An even-dimensional sphere, in particular
S
2
, does not admit a
nonzero vector field.
8.6
Handle Decompositions
This section begins the study of the topological structure of manifolds. We shall now
make good on the promise we made in Section 4.6 to show how closely their struc-
ture is related to critical points on real-valued functions defined on them.
Before we get started, it is important that the reader understand certain notation
used in this section. If an n-dimensional manifold
M
is a subset of
R
m
, then the points
of
M
are m-tuples, but this is the wrong way to look at them. Every point
p
of
M
has
an open neighborhood
V
and a local parameterization j :
U
Æ
V
defined on an open
subset
U
of
R
n
. The function j(u
1
,u
2
,...,u
n
) defines a curvilinear coordinate system
in
V
, so that, typically
(1) one thinks of the point
p
as having coordinates u
i
, and
(2) when one deals with a function f defined on
M
one thinks of f restricted to
V
as a function of the parameters u
i
.
Formally, point (2) means that instead of working with the function f in a neighbor-
hood of the point
p
, one works with the function f
j
(u
1
,u
2
,...,u
n
) = f(j(u
1
,u
2
,...,u
n
)).
Because expressions would become cumbersome if one were to use the precise nota-
tion f
j
, one is sloppy and writes things like f(u
1
,u
2
,...,u
n
) in this case. There should
be no confusion now that we have explained what is meant.
Notation.
Expressions such as “
in local coordinates
u
i
the function f has the form
f(u
1
,u
2
,...,u
n
) = . . .” will mean “f
j
(u
1
,u
2
,...,u
n
) = ...”
This way of talking about functions on manifolds is very common. Of course, there
are many local parameterizations j for
p
, but it will not matter which we choose for
what we want to do, so that we will not bother to mention j explicitly. (In other
contexts, if things do depend on j, then one has to take that dependence into account.)
