Nonlinear Equations - UFRJ

[SEC. 3.2: BROUWER DEGREE 37
to q. Each point of γ(t) admits an arc-length parameterization for
a neighborhood of it. As the path is compact, we can pick a finite
subcovering of those neighborhoods.
By patching together the parameterizations, we obtain one by arc
length X ′ : (a ′ , b ′ ) → M with X ′ (a ′ ) = p, X ′ (b ′ ) = q.
Step 3: Two parameterizations by arc length with X(0) = Y (0)
are equal in the overlap of their domains, or differ by time reversal.
Step 4: Let p ∈ M be an arbitrary interior point. Then, let
X : W → M be the maximal parameterization by arc length with
X(0) → M. The domain W is connected. Now we distinguish two
cases.
Step 4, case 1: X is injective. In that case, X is a diffeomorphism
between M and a connected subset of R
Step 4, case 2: Let r have minimal modulus so that X(0) =
X(r). Unicity of the path-length parameterization implies that for
all k ∈ Z, X(kr) = X(r). In that case, X is a diffeomorphism of the
topological circle R mod r into M.
Exercise 3.1. Give an example of embedded manifold in R n that is
not the preimage of a regular value of a function. (This does not
mean it cannot be embedded into some R N !).
3.2 Brouwer degree
Through this section, let B be an open ball in R n , B denotes its
topological closure, and ∂B its boundary.
Lemma 3.5. Let f : B → R n be a smooth map, extending to a C 1
map ¯f from B to R n . Let Y f ⊂ R n be the set of regular values of
f, not in f(∂B). Then, Y f has full measure and any y ∈ Y f has at
most finitely many preimages in B.
Proof. By Sard’s theorem, the set of regular values of f has full measure.
Moreover, ∂B has finite volume, hence it can be covered by
a finite union of balls of arbitrarily small total volume. Its image
f(∂B) is contained in the image of this union of balls. Since f is
C 1 on B, we can make the volume of the image of the union of balls
arbitrarily small. Hence, f(∂B) has zero measure. Therefore, Y f has
full measure.