Differential Topology - Computer Graphics and Geometric Modeling - page 528

Definition. Let M be a manifold with boundary ∂ M . A collar for ∂ M is an

imbedding

[

) Æ f: ∂¥• M 0, M

such that f( p ,0) = p for all p Œ∂ M . The map f|(∂ M ¥ [0,1]) is called a closed collar for
∂ M . Again, the subsets of M that are the images of the maps are sometimes called a
collar and closed collar, respectively, but one still assumes that the product structure
via a map f is known.

Figure 8.34(b) shows an example of a closed collar.

8.10.13. Theorem. Every boundary of a manifold has a collar.

Proof. See [Hirs76].

We shall use the last theorem to finish this section by showing that if ( M n ,s) is an
oriented n-manifold with boundary, then the orientation s on M induces a unique ori-
entation m on the boundary ∂ M . Let p Œ∂ M and consider T p (∂ M ) as a subset of T p ( M ).
Let n p be a nonzero tangent vector for M that points into M and choose a basis v 1 ,
v 2 ,..., v n-1 for T p (∂ M ) so that [ v 1 , v 2 ,..., v n-1 , n p ] defines the same orientation of T p ( M )
as s( p ). (With a Riemannian metric we could have chosen the vector n p to be normal
to ∂ M .) It is easy to show that the orientation m( p ) = [ v 1 , v 2 ,..., v n-1 ] of T p (∂ M ) is well-
defined. Finally, the fact that the boundary of M has a collar allows us to choose the
vectors n p in such a way that they vary continuously with p , so that the function m
really does define an orientation of ∂ M .

Definition. The orientation m for ∂ M is called the orientation of ∂ M induced by the
orientation s of M .

8.11 Transversality

If there is one concept that is key in the study of manifolds, it is the concept of trans-
versality. We have not run into it much because so many proofs have been omitted in
this chapter. Starting with Theorem 8.6.1, which is essential in understanding the
structure of manifolds, one would find over and over again that proofs need functions
whose singularities are well-structured. The existence of such functions or the ability
to deform a given function into one of that type is what transversality is all about.
This section will attempt to give a brief overview of some basic definitions and prin-
cipal results. The material was not presented earlier because we wanted to discuss it
in the context of abstract manifolds.
In Section 4.8 we defined what it means for a set in R n

to have measure 0. We

extend this concept to subsets of manifolds.

Definition. Let M n be a manifold. A subset A of M is said to have measure
zero if for all coordinate neighborhoods ( U ,j) of M , j( U « A ) has measure zero in R n .