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 $\partial M$. $\Box$