Definition (boundary):
Let
be a differentiable manifold equipped with an atlas
. Further, let
be the set of all
such that
maps to an open subset of the half-space
equipped with its subspace topology w.r.t.
. The boundary of
, commonly denoted by
, is defined as follows:
![{\displaystyle \partial M:=\bigcup _{\beta \in B}\varphi _{\beta }^{-1}\left(\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8975ff532399cc05703ede622459f8aa2e117c)
Proposition (the boundary of a differentiable manifold with boundary is a differentiable manifold):
Let
be a differentiable manifold with boundary of class
and let
be an atlas of
. Then
is a differentiable manifold with boundary of class
, and the family
![{\displaystyle (\pi _{2,\ldots ,n}\circ \varphi _{\alpha }\upharpoonright \partial M)_{\alpha \in A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fba721e3c9eab752dabe4499b9bbc91958d699fd)
constitutes an atlas of
, where
is defined as follows:
![{\displaystyle \pi _{2,\ldots ,n}:\mathbb {R} ^{n}\to \mathbb {R} ^{n-1},~\pi _{2,\ldots ,n}(x_{1},\ldots ,x_{n}):=(x_{2},\ldots ,x_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/494fd803e92fc803a66594494052c201963de572)
Proof: First, we prove that for each
, the function
is a homeomorphism.
To this end, it is prudent to observe that whenever
and
such that
contains
(where
shall denote the domain of definition of
), then
. This is because by the definition of
, there exists a
and an
such that
;
yet the function
is a homeomorphism, whence so is its inverse, so that upon assuming that
, the closedness of the latter set permits the choice of an open neighbourhood
of
that does not intersect
, and Brouwer's invariance of domain theorem then implies that
![{\displaystyle (\varphi _{\alpha }\circ \varphi _{\beta }^{-1})^{-1}(V)=\varphi _{\beta }\circ \varphi _{\alpha }^{-1}(V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/097a2e2a19c258e4a4e5bd721c38b514b029c822)
is an open neighbourhood of
with respect to the Euclidean topology of
, whereas the same set must be contained within the image of
, which is in turn contained within
, so that
cannot intersect
, for otherwise it would contain one of its boundary points and hence be not closed, contradicting the assumption that
.
This proves that whenever
, the function
maps
to
. Hence, when restricted to the image of
, the function
is invertible and in fact a homeomorphism between a subset of
endowed with its subspace topology and
. In fact, restricted in this way,
is a diffeomorphism of class
.
Moreover,
is a homeomorphism since the restriction of a homeomorphism is again a homeomorphism. Hence,
![{\displaystyle \pi _{2,\ldots ,n}\circ \varphi _{\alpha }\upharpoonright \partial M=\pi _{2,\ldots ,n}\upharpoonright \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}\circ \varphi _{\alpha }\upharpoonright \partial M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6184956fecdd103c4d0025ad71cf0fff46361fad)
is a homeomorphism as the composition of homeomorphisms; indeed,
is a homeomorphism between a subset of
and a subset of
.
Let now
. Then
,
and the differentiability condition now follows from the fact that the composition of the three functions
,
and
[[is
times differentiable as the composition of
times differentiable functions]].
Finally, by the very definition of
the domains of definition of the functions in the family
cover all of
.