Nonlinear Equations - UFRJ

36 [CH. 3: TOPOLOGY AND ZERO COUNTING
of charts X α : U α → M covering M, such that whenever V α ∩V β ≠ ∅,
det(D ( )
Xα −1 X β x ) > 0 for all x ∈ U β ∩ X −1
β
(V α).
An orientation of M defines orientations in each T p M. A manifold
admitting an orientation is said to be orientable. If M is orientable
and connected, an orientation in one T p M defines an orientation in
all M.
A 0-dimensional manifold is just a union of disjoint points. An
An orientation for a zero-manifold is an assignment of ±1 to each
point.
If M is an oriented manifold and ∂M is non-empty, the boundary
∂M is oriented by the following rule: let p ∈ ∂M and assume a
parameterization X : U ∩ H m − → M. With this convention we choose
the sign so that u = ± ∂X
∂x n
is an outward pointing vector. We say
that X |U∩[xm=0] is positively oriented if and only if X is positively
oriented.
The following result will be used:
Proposition 3.4. A smooth connected 1-dimensional manifold (possibly)
with boundary is diffeomorphic either to the circle S 1 or to a
connected subset of R.
Proof. A parameterization by arc-length is a parameterization X :
U → M with
∥ ∥∥∥ ∂X
∂x 1
∥ ∥∥∥
= 1.
Step 1: For each interior point p ∈ M, there is a parameterization
X : U → V ∈ M by arc-length.
Indeed, we know that there is a parameterization Y : (a, b) →
V ∋ p, Y (0) = p.
For each q = Y (c) ∈ V , let
{ ∫ c
t(q) =
0 ‖Y ′ (t)‖dt if c ≥ 0
− ∫ 0
c ‖Y ′ (t)‖dt if c ≤ 0
The map t : V → R is a diffeomorphism of V into some interval
(d, e) ⊂ R. Let U = (d, e) and X = Y ◦ t −1 . Then X : U → M is a
parameterization by arc length.
Step 2: Let p be a fixed interior point of M. Let q be an arbitrary
point of M. Because M is connected, there is a path γ(t) linking p