Nonlinear Equations - UFRJ

[SEC. 3.1: MANIFOLDS 35
Note that if X : U → M and Y : V → M are two local charts
and domains X(U) ∩ Y (V ) ≠ ∅, then Y −1 ◦ X is a diffeomorphism,
of the same class as Φ.
A smooth (resp. C k , resp. analytic) m-dimensional abstract manifold
is a topological space M such that, for every p ∈ M, there is a
neighborhood of p in M that is smoothly (resp. C k , resp. analytically)
diffeomorphic to an embedded m-dimensional manifold of the same
differentiability class. Whitney’s embedding theorem guarantees that
a smooth abstract m-dimensional manifold can be embedded in R 2m .
Let H m + (resp.) H m − be the closed half-space in R m defined by the
inequation x m ≥ 0 (resp. x m ≤ 0).
Definition 3.3 (Embedded manifold with boundary). A smooth
(resp. C k for k ≥ 1, resp. analytic) m-dimensional real manifold
M with boundary, embedded in R n is a subset M ⊆ R n with the following
property: for any p ∈ M, there are open sets U ⊆ H m + or H m − ,
p ∈ V ⊆ R n , and a smooth (resp. C k , resp. analytic) diffeomorphism
X : U → M ∩ V . The map X is called a parameterization or a chart.
The boundary ∂M of an embedded manifold M is the union of
the images of the X(U ∩ [x m = 0]). It is also a smooth (resp. C k
resp. analytic) manifold (without boundary) of dimension m − 1.
Note the linguistic trap: every manifold is a manifold with boundary,
while a manifold with boundary does not need to have a nonempty
boundary.
Let E be a finite-dimensional real linear space. We say that two
bases (α 1 , . . . , α m ) and (β 1 , . . . , β m ) of E have the same orientation
if and only if det A > 0, where A is the matrix relating those two
bases:
α i = ∑ A ij β j .
j
There are two possible orientations for a linear space. The canonical
orientation of R m is given by the canonical basis (e 1 , . . . , e m ).
The tangent space of M at p, denoted by T p M, is the image of
DX p ⊆ R n . An orientation for an m-dimensional manifold M with
boundary (this includes ordinary manifolds !) when m ≥ 1 is a class