Manifolds are sets which resemble ℝ^{n} locally. To make the concept of a manifold more
precise, here is a definition.
Definition 28.1.1Let Ω ⊆ ℝ^{m}. A set, U, is open in Ω if it is the intersection ofan open set from ℝ^{m}with Ω. Equivalently, a set, U is open in Ω if for every point,x ∈ U, there exists δ > 0 such that if
|x− y |
< δ and y ∈ Ω, then y ∈ U. A set, H,is closed in Ω if it is the intersection of a closed set from ℝ^{m}with Ω. Equivalently,a set, H, is closed in Ω if whenever, y is a limit point of H and y ∈ Ω, it followsy ∈ H.
Recall the following definition.
Definition 28.1.2Let V ⊆ ℝ^{n}. C^{k}
(-- m )
V;ℝ
is the set of functions which arerestrictions to V of some function defined on ℝ^{n}which has k continuous derivativesand compact support. When k = 0, it means the restriction to V of continuousfunctions with compact support.
Definition 28.1.3A closed and bounded subset of ℝ^{m}, Ω, will be called an ndimensional manifold with boundary, n ≥ 1, if there are finitely many sets, U_{i},open in Ω and continuous one to one functions, R_{i}∈ C^{0}
(- n)
Ui,ℝ
such thatR_{i}U_{i}is relatively open in ℝ_{≤}^{n}≡
n
{u ∈ ℝ : u1 ≤ 0}
, R_{i}^{−1}is continuous. Thesemappings, R_{i}, together with their domains, U_{i}, are called charts and the totalityof all the charts,
(Ui,Ri)
just described is called an atlas for the manifold.Define
n
int(Ω) ≡ {x ∈ Ω : for some i,Rix ∈ ℝ< }
where ℝ_{<}^{n}≡
{u ∈ ℝn : u1 < 0}
. Also define
n
∂Ω ≡ {x ∈ Ω : for some i,Rix ∈ ℝ0}
where
ℝn0 ≡ {u ∈ ℝn : u1 = 0}
and ∂Ω is called the boundary of Ω. Note that if n = 1, ℝ_{0}^{n}is just the single point 0. Byconvention, we will consider the boundary of such a 0 dimensional manifold to beempty.
This definition is a little too restrictive. In general the collection of sets, U_{i} is not
finite. However, in the case where Ω is closed and bounded, compactness of Ω can be
used to get a finite covering and since this is the case of most interest here, the
assumption that the collection of sets, U_{i}, is finite is made. However, most of
what is presented here can be generalized to the case of a locally finite atlas.
Theorem 28.1.4Let ∂Ω andint
(Ω )
be as defined above. Thenint
(Ω )
is openin Ω and ∂Ω is closed in Ω. Furthermore, ∂Ω ∩int
(Ω)
= ∅, Ω = ∂Ω ∪int
(Ω)
, andfor n ≥ 2, ∂Ω is an n−1 dimensional manifold for which ∂
(∂Ω)
= ∅. The propertyof being inint
(Ω)
or ∂Ω does not depend on the choice of atlas.
Proof: It is clear that Ω = ∂Ω ∪int
(Ω )
. First consider the claim that ∂Ω ∩int
(Ω )
= ∅.
Suppose this does not happen. Then there would exist x ∈ ∂Ω ∩int
(Ω)
. Therefore, there
would exist two mappings R_{i} and R_{j} such that R_{j}x ∈ ℝ_{0}^{n} and R_{i}x ∈ ℝ_{<}^{n} with
x ∈ U_{i}∩U_{j}. Now consider the map, R_{j}∘ R_{i}^{−1}, a continuous one to one map from ℝ_{≤}^{n}
to ℝ_{≤}^{n} having a continuous inverse. By continuity, there exists r > 0 small enough
that,
R−i1B (Rix,r) ⊆ Ui ∩ Uj.
Therefore, R_{j}∘R_{i}^{−1}
(B (R x,r))
i
⊆ ℝ_{≤}^{n} and contains a point on ℝ_{0}^{n},R_{j}x. However, this
cannot occur because it contradicts the theorem on invariance of domain, Theorem
21.5.3, which requires that R_{j}∘R_{i}^{−1}
(B (R x,r))
i
must be an open subset of ℝ^{n} and this
one isn’t because of the point on ℝ_{0}^{n}. Therefore, ∂Ω ∩int
(Ω)
= ∅ as claimed. This same
argument shows that the property of being in int
(Ω)
or ∂Ω does not depend on the
choice of the atlas.
To verify that ∂
(∂Ω)
= ∅, let S_{i} be the restriction of R_{i} to ∂Ω ∩ U_{i}. Thus
Si(x ) = (0,(Rix ) ,⋅⋅⋅,(Rix) )
2 n
and the collection of such points for x ∈ ∂Ω ∩ U_{i} is an open bounded subset
of
{u ∈ ℝn : u1 = 0},
identified with ℝ^{n−1}. S_{i}
(∂Ω ∩ U )
i
is bounded because S_{i} is the restriction of a continuous
function defined on ℝ^{m} and ∂Ω ∩U_{i}≡ V_{i} is contained in the compact set Ω. Thus if S_{i} is
modified slightly, to be of the form
′
Si(x) = ((Rix )2 − ki,⋅⋅⋅,(Rix )n)
where k_{i} is chosen sufficiently large enough that
(Ri (Vi))
_{2}− k_{i}< 0, it follows that
{(Vi,S′i)}
is an atlas for ∂Ω as an n− 1 dimensional manifold such that every point of ∂Ω
is sent to to ℝ_{<}^{n−1} and none gets sent to ℝ_{0}^{n−1}. It follows ∂Ω is an n − 1 dimensional
manifold with empty boundary. In case n = 1, the result follows by definition of the
boundary of a 0 dimensional manifold.
Next consider the claim that int
(Ω )
is open in Ω. If x ∈int
(Ω)
, are all points of Ω
which are sufficiently close to x also in int
(Ω)
? If this were not true, there would exist
{xn}
such that x_{n}∈ ∂Ω and x_{n}→ x. Since there are only finitely many charts of
interest, this would imply the existence of a subsequence, still denoted by x_{n} and
a single map, R_{i} such that R_{i}
(xn)
∈ ℝ_{0}^{n}. But then R_{i}
(xn )
→ R_{i}
(x)
and
so R_{i}
(x)
∈ ℝ_{0}^{n} showing x ∈ ∂Ω, a contradiction to int
(Ω )
∩ ∂Ω = ∅. Now
it follows that ∂Ω is closed in Ω because ∂Ω = Ω ∖int
(Ω )
. This proves the
Theorem.
Definition 28.1.5An n dimensional manifold with boundary, Ω is a C^{k}manifold withboundary for some k ≥ 1 if
( )
Rj ∘R −1∈ Ck Ri(Ui-∩Uj);ℝn
i
and R_{i}^{−1}∈ C^{k}
(----- )
RiUi;ℝm
. It is called a continuous or Lipschitz manifold withboundary if the mappings, R_{j}∘ R_{i}^{−1}, R_{i}^{−1},R_{i}are respectively continuous orLipschitz continuous. In the case where Ω is a C^{k},k ≥ 1 manifold, it is calledorientable if in addition to this there exists an atlas,
(Ur,Rr)
, such that wheneverU_{i}∩ U_{j}≠∅,
( ( −1))
det D Rj ∘R i (u) > 0 for all u ∈ Ri (Ui ∩ Uj) (28.1.1)
(28.1.1)
The mappings, R_{i}∘R_{j}^{−1}are called the overlap maps. In the case where k = 0, the R_{i}areonly assumed continuous so there is no differentiability available and in thiscase, the manifold is oriented if whenever A is an open connected subset ofint
(Ri (Ui ∩Uj ))
whose boundary has measure zero and separates ℝ^{n}into twocomponents,
( −1)
d y,A,Rj ∘ Ri ∈ {1,0} (28.1.2)
(28.1.2)
depending on whether y ∈ R_{j}∘ R_{i}^{−1}
(A)
. An atlas satisfying 28.1.1or more generally28.1.2is called an oriented atlas.
It follows from Proposition 21.6.5 the degree in 28.1.2 is either undefined if
y ∈ R_{j}∘ R_{i}^{−1}∂A or it is 1, -1,or 0.
The study of manifolds is really a generalization of something with which everyone
who has taken a normal calculus course is familiar. We think of a point in three
dimensional space in two ways. There is a geometric point and there are coordinates
associated with this point. There are many different coordinate systems which describe a
point. There are spherical coordinates, cylindrical coordinates and rectangular
coordinates to name the three most popular coordinate systems. These coordinates are
like the vector u. The point, x is like the geometric point although it is always assumed x
has rectangular coordinates in ℝ^{m} for some m. Under fairly general conditions, it can be
shown there is no loss of generality in making such an assumption. Next is some
algebra.