Differential Topology - Computer Graphics and Geometric Modeling - page 535

number of t M is the intersection number I(D M ,D M ) of the diagonal D M with itself in
M ¥ M .

8.11.13. Theorem. The Euler number of the tangent bundle of a closed compact
oriented manifold M is just the Euler characteristic of M , that is, I(t M ) = c( M ).

Proof. See [Miln65b] and [MilS74]. Let s 0 be the zero cross-section of t M . Because
the proof of Theorem 8.11.11 would show that I(t M ) is determined from the inter-
sections of s( M ) and s 0 ( M ) for any cross-section s of t M , a major part of the proof of
this theorem involves finding an s for which one can relate the indices I q (s,s 0 ( M )) at
a point q where s( M ) and s 0 ( M ) intersect to the Euler characteristic.
Assume that M has a Riemannian metric <,> and let f : M Æ R be a function. Con-
sider the gradient vector field —f. This vector field vanishes at precisely the critical
points of f. Assume that f has only nondegenerate critical points. One can show that
at a critical point p of index k, I s( p ) (s,s 0 ( M )) = (-1) k . This fact and Theorem 8.6.5 proves
what we want to show.

Theorem 8.11.13 explains why I(x) is called the “Euler” number of the vector
bundle x.
We finish this section by stating one more general result about transversality.

8.11.14. Theorem. Let M and N be differentiable manifolds and let A be a sub-
manifold of N . The set of differentiable maps f : M Æ N that are transverse to A is
dense in the space of all differentiable maps f : M Æ N (the latter space can be given
a natural topology that basically says that functions are close if their derivatives are
close).

Proof. See [Hirs76].

Theorem 8.11.14 is a really fundamental theorem. It is just one of many theorems
of that type. In effect, these theorems say that maps between manifolds are always
close to and homotopic to maps that satisfy an appropriate transversality property, so
that there is no loss in generality if we assume that the original map has the desired
transversality property.

8.12 Differential Forms and Integration

In Section 4.8 we defined the integral of real-valued functions defined on subsets of
R n . Section 4.9.1 extended the theory to integrating differential forms on open subsets
of R n . This section will sketch how one can integrate over manifolds. In the case of
submanifolds of R n , given a real-valued function f defined on the manifold, we could
of course approximate the manifold by a simplicial complex or other polygonal man-
ifold and then use a Riemann type sum of the form

Â

()( )

volume s f p ,

s

polygonal cells s