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The really different differentiable structures are rather exotic. Also, when one builds
new manifolds from old ones, the constructions invariably induce a natural differen-
tiable structure on the new manifolds from the differentiable structure on the old
ones. Finally and most importantly, the actual differentiable structure usually does
not matter, only that one has one.
The reader may be wondering how differentiable manifolds differ from topologi-
cal manifolds. A deep result in the theory of manifolds says that not all topological
manifolds admit a differentiable structure. Now, the boundary of a square is a topo-
logical manifold that is not a differentiable submanifold of
R
n
, but being homeo-
morphic to
S
1
, it
does
admit a nice differentiable structure. Therefore, saying that a
manifold does not admit a differentiable structure is saying much more, namely, that
every
imbedding of it in
R
n
has “corners.” It follows that there is a big difference
between C
0
and C
1
manifolds, but it turns out that there is essentially no difference
between C
1
and C
•
manifolds (see [Munk61] or [Hirs76]). For this reason and in order
not to get bogged down with technical issues as to how much differentiability one
needs for a result, we mostly consider C
•
manifolds. Also, when it comes to maps, the
next theorem shows that we can basically assume that all maps between differentiable
manifolds are differentiable.
8.8.5. Theorem.
Any continuous map between differentiable manifolds can be
“approximated” by a differentiable map that is homotopic to the original. Further-
more, any two homotopic differentiable maps are homotopic by a differentiable
homotopy.
Proof.
See [Hirs76] for a precise statement and proofs.
Proving the results we just mentioned about differentiable structures is beyond
the scope of this topic. See Notes 1- 4 in Section 6.5 for a few other related comments.
It follows from Theorem 4.4.5 that locally the imbedding of a k-dimensional smooth
manifold
M
k
in
R
n
looks like the imbedding of
R
k
in
R
n
. The next two theorems relate
the manifolds defined in Section 8.3 to the abstract manifolds we are studying now.
Combined, the theorems say that there is no difference between the two types of
spaces. It is simply a case of two different ways of looking at the same thing.
8.8.6. Theorem.
A subset of
R
n
is a differentiable manifold in the sense of Section
8.3 if and only if it is a differentiable submanifold of
R
n
in the sense of this section.
Proof.
This is a consequence of the Inverse Function Theorem.
Every abstract manifold
M
k
8.8.7. Theorem.
can be imbedded in some Euclidean
space
R
n
.
Proof.
See [Miln58], [Munk61], or [Hirs76].
The proof of Theorem 8.8.7 is not very hard if one is happy with any n. A classi-
cal result of H. Whitney, proved in 1936, states that n = 2k + 1 suffices. In fact, one
can improve this to n = 2k, but this is even more difficult.
