# Differentiable Manifolds/The definition of differentiable manifolds

Definition (differentiable manifold):

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

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

satisfying the following conditions:

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

Definition (atlas):

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

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

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

Definition (chart):

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

Definition (boundary):

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

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

Definition (differentiable manifold with boundary):

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

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

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

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

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

${\displaystyle \pi _{2,\ldots ,n}:\mathbb {R} ^{n}\to \mathbb {R} ^{n-1},~\pi _{2,\ldots ,n}(x_{1},\ldots ,x_{n}):=(x_{2},\ldots ,x_{n})}$

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

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

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

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

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

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

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

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

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

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

Let now ${\displaystyle \alpha ,\beta \in A}$. Then

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

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

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