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In-Depth Information
CHAPTER 8
Differential Topology
8.1
Introduction
Most of what we did in the last three chapters applied to topological spaces that could
be quite general, even if one restricted oneself to polyhedra. In this chapter we spe-
cialize to studying manifolds. Topological manifolds were defined in Chapter 5 and
then studied further in the context of pseudomanifolds and homology manifolds in
Chapter 7. As topological spaces they look like
R
n
or
R
+
locally. To put it another way,
to a sufficiently small bug in a manifold the space around it would look flat. Now we
shall study differentiable manifolds, which have a differentiable structure in addition
to their topology. Having a differentiable structure on a manifold means that we can
transfer many other properties and techniques from Enclidean space over to it. In
particular, we can use calculus and linear algebra. This turns out to play an im-
portant role in the analysis of the manifold because we will be able to do things that
we can not do with an ordinary topological manifold. Smooth curves and surfaces,
spaces important in geometric modeling and CAGD, are instances of differentiable
manifolds.
Although we sketch an intrinsic definition of a differentiable manifold later in this
chapter, our working definition will be one that defines certain subspaces of
R
n
to be
manifolds. By defining a manifold in terms of what could be considered as a partic-
ular imbedding of the corresponding “abstract” manifold, we simplify the definition
of tangent vector and tangent space, which are an essential part of a manifold. The
intrinsic definition of a manifold is not that much harder, but with our approach we
make it somewhat easier for the reader who has never seen any of this material before
and who does not feel entirely comfortable with n dimensions. Defining manifolds as
parameterized subspaces of Euclidean space is also the way one usually sees them
defined in CAGD. In one sense, there is no loss of generality because an important
theorem asserts that even abstract manifolds can be realized as subspaces of a suit-
ably high dimensional Euclidean space. On the other hand, one should be aware of
the fact that the disadvantage of studying manifolds as subsets of Euclidean space is
that it might seem as if some of the invariants we define for them depend on the sur-
rounding space when in fact they do not. Abstraction enables one to see essential
