Differentiable Manifolds/The definition of differentiable manifolds

Definition (differentiable manifold):

Let $k\in \mathbb {N}$ . Then a differentiable manifold of class ${\mathcal {C}}^{k}$ is a topological space $M$ together with a family of functions $(\varphi _{\alpha })_{\alpha \in A}$ , such that each $\varphi _{\alpha }$ is a homeomorphism defined on an open subset $U_{\alpha }\subseteq M$ whose image is

either an open subset of $\mathbb {R} ^{n}$
or an open subset of the half-space $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}\geq 0\}$ with respect to the subspace topology

satisfying the following conditions:

For all $\alpha ,\beta \in A$ , the function $\varphi _{\alpha }\circ \varphi _{\beta }^{-1}$ is $k$ times continuously differentiable on its domain of definition
For all $p\in M$ there exists an $\alpha \in A$ such that $p\in U_{\alpha }$

Definition (atlas):

Let $M$ be a differentiable manifold of class ${\mathcal {C}}^{k}$ that is defined using a family $(\varphi _{\alpha })_{\alpha \in A}$ of functions. An atlas of $M$ is a family $(\psi _{\beta })_{\beta \in B}$ , such that each $\psi _{\beta }$ is a homeomorphism defined on an open subset $V_{\beta }\subseteq M$ whose image is

either an open subset of $\mathbb {R} ^{n}$
or an open subset of the half-space $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}\geq 0\}$ with respect to the subspace topology

so that the family $(\psi _{\beta })_{\beta \in B}$ is compatible with the family $(\varphi _{\alpha })_{\alpha \in A}$ in the sense that for all $\beta \in B$ and $\alpha \in A$ , the two functions $\varphi _{\alpha }\circ \psi _{\beta }^{-1}$ and its inverse $\psi _{\beta }\circ \varphi _{\alpha }^{-1}$ are $k$ times continuously differentiable on their respective domains of definition, and so that for each $p\in M$ there exists a $\beta \in B$ such that $p\in V_{\beta }$ .

Definition (chart):

Let $M$ be a differentiable manifold. Then a chart of $M$ is a function $\varphi _{\alpha }$ for some $\alpha \in A$ , where $(\varphi _{\alpha })_{\alpha \in A}$ is any atlas of $M$ .

Definition (boundary):

Let $M$ be a differentiable manifold equipped with an atlas $(\varphi _{\alpha })_{\alpha \in A}$ . Further, let $B\subseteq A$ be the set of all $\alpha \in A$ such that $\varphi _{\alpha }$ maps to an open subset of the half-space $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}\geq 0\}$ equipped with its subspace topology w.r.t. $\mathbb {R} ^{n}$ . The boundary of $M$ , commonly denoted by $\partial M$ , is defined as follows:

\partial M:=\bigcup _{\beta \in B}\varphi _{\beta }^{-1}\left(\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}\right)

Definition (differentiable manifold with boundary):

A differentiable manifold with boundary is a differentiable manifold $M$ equipped with an atlas $(\varphi _{\alpha })_{\alpha \in A}$ such that the image of at least one chart $\varphi _{\alpha }$ is an open subset of $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}\geq 0\}$ (equipped with its subspace topology w.r.t. $\mathbb {R} ^{n}$ ) that intersects the boundary set $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ .

Proposition (the boundary of a differentiable manifold with boundary is a differentiable manifold):

Let $M$ be a differentiable manifold with boundary of class ${\mathcal {C}}^{k}$ and let $(\varphi _{\alpha })_{\alpha \in A}$ be an atlas of $M$ . Then $\partial M$ is a differentiable manifold with boundary of class ${\mathcal {C}}^{k}$ , and the family

(\pi _{2,\ldots ,n}\circ \varphi _{\alpha }\upharpoonright \partial M)_{\alpha \in A}

constitutes an atlas of $\partial M$ , where $\pi _{2,\ldots ,n}$ is defined as follows:

\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})

Proof: First, we prove that for each $\alpha \in A$ , the function $\pi _{2,\ldots ,n}\circ \varphi _{\alpha }\upharpoonright \partial M$ is a homeomorphism.

To this end, it is prudent to observe that whenever $p\in \partial M$ and $\alpha \in A$ such that $U_{\alpha }$ contains $p$ (where $U_{\beta }\subseteq M$ shall denote the domain of definition of $\varphi _{\alpha }$ ), then $\varphi _{\alpha }(p)\in \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ . This is because by the definition of $\partial M$ , there exists a $\beta \in A$ and an $x\in \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ such that

p=\varphi _{\beta }^{-1}(x)

;

yet the function $\varphi _{\alpha }\circ \varphi _{\beta }^{-1}$ is a homeomorphism, whence so is its inverse, so that upon assuming that $\varphi _{\alpha }(p)=\varphi _{\alpha }\circ \varphi _{\beta }^{-1}(x)\notin \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ , the closedness of the latter set permits the choice of an open neighbourhood $V$ of $\varphi _{\alpha }(p)$ that does not intersect $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ , and Brouwer's invariance of domain theorem then implies that

(\varphi _{\alpha }\circ \varphi _{\beta }^{-1})^{-1}(V)=\varphi _{\beta }\circ \varphi _{\alpha }^{-1}(V)

is an open neighbourhood of $x$ with respect to the Euclidean topology of $\mathbb {R} ^{n}$ , whereas the same set must be contained within the image of $\varphi _{\beta }$ , which is in turn contained within $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}\geq 0\}$ , so that $\varphi _{\beta }\circ \varphi _{\alpha }^{-1}(V)$ cannot intersect $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ , for otherwise it would contain one of its boundary points and hence be not closed, contradicting the assumption that $\varphi _{\beta }(p)\in \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ .

This proves that whenever $\alpha \in A$ , the function $\varphi _{\alpha }$ maps $\partial M$ to $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ . Hence, when restricted to the image of $\varphi _{\alpha }\upharpoonright \partial M$ , the function $\pi _{2,\ldots ,n}$ is invertible and in fact a homeomorphism between a subset of $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ endowed with its subspace topology and $\mathbb {R} ^{n}$ . In fact, restricted in this way, $\pi _{2,\ldots ,n}$ is a diffeomorphism of class ${\mathcal {C}}^{\infty }$ .

Moreover, $\varphi _{\alpha }\upharpoonright \partial M$ is a homeomorphism since the restriction of a homeomorphism is again a homeomorphism. Hence,

\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

is a homeomorphism as the composition of homeomorphisms; indeed, $\varphi _{\alpha }\upharpoonright \partial M$ is a homeomorphism between a subset of $\partial M$ and a subset of $\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}$ .

Let now $\alpha ,\beta \in A$ . Then

\pi _{2,\ldots ,n}\circ \varphi _{\alpha }\upharpoonright \partial M\circ (\pi _{2,\ldots ,n}\circ \varphi _{\beta }\upharpoonright \partial M)^{-1}=\pi _{2,\ldots ,n}\circ (\varphi _{\alpha }\circ \varphi _{\beta }^{-1})\circ (\pi _{2,\ldots ,n}\upharpoonright \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\})^{-1}

,

and the differentiability condition now follows from the fact that the composition of the three functions $\pi _{2,\ldots ,n}$ , $\varphi _{\alpha }\circ \varphi _{\beta }^{-1}$ and $(\pi _{2,\ldots ,n}\upharpoonright \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\})^{-1}$ [[is $k$ times differentiable as the composition of $k$ times differentiable functions]].

Finally, by the very definition of $\partial M$ the domains of definition of the functions in the family $(\pi _{2,\ldots ,n}\circ \varphi _{\alpha }\upharpoonright \partial M)_{\alpha \in A}$ cover all of $\partial M$ . $\Box$

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