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aspects more clearly. That is the reason that one defines an abstract n-dimensional
vector space and does not just deal with
R
n
(although the two are isomorphic). In the
case of manifolds, they have lots of intrinsic properties that do not depend on any
particular imbedding. For example, on one level anyway, there really is no difference
between all the circles of radius one in the plane. They all correspond to different
imbeddings of an “abstract” version of a unit circle. In fact, until we get to differen-
tial geometry where the metric matters,
all
circles are the “same.”
Section 8.2 discusses parameterizations of spaces, which is essential for the defi-
nition of a manifold given in Section 8.3 and the abstract manifolds defined in Section
8.8. The idea of a manifold originated in Riemann's groundbreaking lecture “On the
Hypotheses which lie at the Foundation of Geometry” delivered to the faculty at the
University of Göttingen in 1854. The ideas expressed in this talk are usually consid-
ered to be the most influential in the history of differential geometry. An integral part
of a differentiable manifold is its tangent space, which is defined in Section 8.4.
Section 8.5 discusses what it means for a manifold to be orientable. Sections 8.6 and
8.7 give an overview of what is involved in the classification of manifolds. They give
the reader a taste of some difficult but beautiful results on the structure of manifolds.
Key to this is the handle decomposition of a manifold and cobordism theory along
with algebraic topology invariants. This part of the theory is relatively new. It covers
about a twenty year period starting in the middle 1950s and culminated in the main
structure theorems for manifolds that are known today. See [AgoM76b]. Next, in
Section 8.8 we move on to an intrinsic definition of a manifold. Sections 8.9 and 8.10
define vector bundles and discuss some of their basic properties with emphasis on
their role in the study of manifolds. Section 8.11 defines what it means for maps or
manifolds to be transverse. The degree of a map and intersection numbers serve as
two examples of how transversality can be used, but it appears in a great many essen-
tial ways in proofs related to differentiable manifolds. In Section 8.12 we continue
the topic of differential forms and integration that we started in Sections 4.9 and 4.9.1,
but this time in the setting of manifolds. Finally, in Section 8.13 we take another look
at the projective spaces
P
n
and we introduce the Grassmann manifolds in Section
8.14.
8.2
Parameterizing Spaces
Parameterizations are a generalization of Descartes' idea that a good approach to
studying a geometric space is to introduce coordinates for its points, because one can
then use equations or other analytic tools to study the space. They are used in many
places and are fundamental to the idea of a manifold, especially the abstract mani-
folds defined at the end of this chapter.
Let
X
be a subset of
R
n
. If there a subset
U
of
R
k
and a surjective C
r
map
Definition.
F:
UX
Æ
,
then F is called a
C
r
parameterization
of
X
. The set
X
is called the
underlying space
of
F in that case.
