Related concepts: topological manifolds, chart, dimension, \({C}^{\infty}\)-compatible, \({C}^{\infty}\) atlas, maximal atlas, smooth manifold
Topological manifolds is a topological space \(\mathcal{M}\) with following properties:
The pair \(\left(G,\phi:G\rightarrow{\mathbb{R}}^{n}\right)\) is a chart, \(n\) is called the dimension of \(\mathcal{M}\).
Two charts \(\left(U,\phi:U\rightarrow{\mathbb{R}}^{n}\right)\) and \(\left(V,\psi:V\rightarrow{\mathbb{R}}^{n}\right)\) are \({C}^{\infty}\)-compatible if both \(\phi\circ{\psi}^{-1}\) and \(\psi\circ{\phi}^{-1}\) are \({C}^{\infty}\) function in \({\mathbb{R}}^{n}\).
A collection \(\mathcal{U}=\left\{\left({U}_{\alpha},{\phi}_{\alpha}\right),\alpha\in\mathcal{A}\right\}\) is a \({C}^{\infty}\) atlas if \(\mathcal{U}\) is pairwise-compatible and an open cover of \(\mathcal{M}\).
\(\mathcal{U}\) is a maximal atlas of \(\mathcal{M}\) if it is not contained in any atlas, i.e. \(\mathcal{U}=\mathcal{V}\) if \(\mathcal{V}\) is any atlas containing \(\mathcal{U}\).
a smooth manifold or manifold is a topological manifold with maximal atlas.
Any atlas on a locally Eulidean space is contained in a unique maximal atlas.
An example: \(\mathbb{R}\) is a manifold under two different charts \(\left(\mathbb{R},\phi\left(t\right)=t\right)\), \(\left(\mathbb{R},\phi\left(t\right)={t}^{3}\right)\), but these two charts are not compatible.
Related concepts: smooth function, smooth map, diffeomorphism, partial derivatives, Jacobian matrix
A function \(f:\mathcal{M}\rightarrow{\mathbb{R}}^{n}\) is smooth at \(p\) if there is a chart \(\left(U,\phi\right)\) about \(p\) such that \(f\circ{\phi}^{-1}\) is smooth at \(\phi\left(p\right)\). \(f\) is smooth on \(M\) if \(f\) is smooth at each point \(p\) on \(\mathcal{M}\).
Let \(\mathcal{M},\mathcal{N}\) be two manifolds, a continuous map \(f:\mathcal{M}\rightarrow\mathcal{N}\) is smooth at \(p\) if there are two charts \(\left(U,\phi\right),\left(V,\psi\right)\) about \(p,f\left(p\right)\) such that \(\psi\circ f\circ{\phi}^{-1}\) is smooth at \(\phi\left(p\right)\). \(f\) is smooth on \(\mathcal{M}\) if \(f\) is smooth at each point of \(\mathcal{M}\).
A diffeomorphism is a bijection \(f:\mathcal{M}\rightarrow\mathcal{N}\) such that \(f\) and \({f}^{-1}\) are both smooth.
Let \(\mathcal{M},\mathcal{N}\) be two manifolds with dimension \(m,n\), a continuous map \(f:\mathcal{M}\rightarrow\mathcal{N}\) is smooth at \(p\), let \(\left(U,\phi\right),\left(V,\psi\right)\) be two charts about \(p\) and \(f\left(p\right)\), a partial derivative at \(p\) is
\[\frac{\partial{f}^{j}}{\partial{x}^{i}}\left(p\right)=\frac{\partial\psi\circ{f}^{j}\circ{\phi}^{-1}}{\partial{r}^{i}}\left(\phi\left(p\right)\right),\quad i=1,\dots,m,\quad j=1,\dots,n,\]the Jacobian matrix is
\[J F\left(p\right)={\left(\frac{\partial{f}^{j}}{\partial{x}^{i}}\left(p\right)\right)}_{i j}.\]Related concepts: germs, derivative, tangent vector, tangent space, differential, chain rule, immersion, submersion, critical point, regular point, critical value, regular value
Let \(\mathcal{M}\) be a manifold, \(p\in\mathcal{M}\), \(f:\mathcal{M}\rightarrow\mathbb{R}\) is smooth, we can define a equivalent class \(f\sim g\Leftrightarrow\exists\text{a neighbourhood }U\text{ of }p\text{, s.t. }f=g\text{ on }U\). The equivalent class are germs. We denote all the equivalent class as \({C}_{p}^{\infty}\left(\mathcal{M}\right)\).
A derivative on a manifold \(\mathcal{M}\) and \(p\in\mathcal{M}\) is a linear functional \(D:{C}_{p}^{\infty}\left(\mathcal{M}\right)\rightarrow\mathbb{R}\) such that
\[D\left(f g\right)=\left(D f\right)g+f\left(D g\right),\]
A tangent vector at \(p\) in \(\mathcal{M}\) is a derivation at p in \(\mathcal{M}\). We denote all the tangent vectors as \({T}_{p}\mathcal{M}\), the tangent space.
Tangent space is a finite dimensional linear space, and the directional derivatives
\[\left\{\frac{\partial}{\partial{x}^{1}},\dots,\frac{\partial}{\partial{x}^{n}}\right\}\]are a basis of \({T}_{p}\mathcal{M}\). This implies that the dimension of manifold is equal to the dimension of its tangent space.
Let \(F\) be a smooth map between \(\mathcal{M}\) and \(\mathcal{N}\). At point \(p\in\mathcal{M}\), differential \({F}_{*}\) is induced by \(F\), \({F}_{*}\) is a linear maps bewteen tangent spaces \({T}_{p}\mathcal{M}\) and \({T}_{F\left(p\right)}\mathcal{N}\):
\[{F}_{*,p}\left({X}_{p}\right)f={X}_{p}\left(f\circ F\right).\]
Let \(\mathcal{M},\mathcal{N}\) be two manifolds with dimension \(m,n\), \(F:\mathcal{M}\rightarrow\mathcal{N}\) is smooth at \(p\), let \(\left(U,\phi\right),\left(V,\psi\right)\) be two charts about \(p\) and \(F\left(p\right)\), the Jacobian matrix of \(F\) is \(J F\left(p\right)={\left(\frac{\partial{f}^{j}}{\partial{x}^{i}}\left(p\right)\right)}_{i j}\), then the matrix of \({F}_{*,p}\) (as a linear map) is \(J F\left(p\right)\).
(Chain Rule) Let \(F:\mathcal{M}\rightarrow\mathcal{N}\), \(G:\mathcal{N}\rightarrow\mathcal{P}\) are smooth, the differential of composition \(G\circ F\) equals the composition of differential
\[{\left(G\circ F\right)}_{*,p}={G}_{*,F\left(p\right)}\circ{F}_{*,p}.\]
Let \(F:\mathcal{N}\rightarrow\mathcal{M}\) is a diffeomorphism at \(p\), then \({F}_{*,p}:{T}_{p}\mathcal{N}\rightarrow{T}_{f\left(p\right)}\mathcal{M}\) is an isomorphism. This implies that if \(F\) is a diffeomorphism, then \(\dim\mathcal{N}=\dim\mathcal{M}\).
Let \(F:\mathcal{N}\rightarrow\mathcal{M}\) be smooth, and \(p\in\mathcal{N}\). \(F\) is an immersion iff \({F}_{*,p}\) is an injective.
Let \(F:\mathcal{N}\rightarrow\mathcal{M}\) be smooth, and \(p\in\mathcal{N}\). \(F\) is a submersion iff \({F}_{*,p}\) is a surjective.
Let \(\dim\mathcal{N}=n\),\(\dim\mathcal{M}=m\) . \(F\) is immersion implies that \(n\leqslant m\), and \(\text{rank}F=n\). \(F\) is submersion implies that \(n\geqslant m\), and \(\text{rank}F=m\).
\(p\in\mathcal{N}\) is a critical point iff \({F}_{*,p}\) is not a surjective.
\(p\in\mathcal{N}\) is a regular point iff \({F}_{*,p}\) is a surjective.
\(q\in\mathcal{M}\) is a critical value iff there exists \(p\in{f}^{-1}\left(q\right)\) such that \(p\) is a critical point.
\(q\in\mathcal{M}\) is a regular value iff \(q\) is not a critical value.
Related concepts: regular submanifold, level set, zero set, regular level set theorem
A subset \(S\) of manifold \(\mathcal{N}\) of dim \(n\) is a regular submanifold of dim \(k\) iff forall \(p\in S\) there exists a chart \(\left(U,\phi\right)\) of \(\mathcal{N}\) suct that \(U\cap S\) is obtained by vanishing of \(n-k\) coordinates of \(\phi\).
| We give the submanifold \(S\) the subspace topology and atlas $$\left{\left(U\cap S,\phi{ | }_{s}\right)\right}$$, then a regular submanifold is a manifold. |
| A level set of \(F:\mathcal{N}\rightarrow\mathcal{M}\) is $${f}^{-1}\left(c\right)=\left{p\in\mathcal{N} | f\left(p\right)=c\right}$$. |
The zero set of \(F:\mathcal{N}\rightarrow\mathcal{M}\) is \({f}^{-1}\left(0\right)\).
The regular level set is the level set of regular value \(c\).
(Regular Level Set) Let \(F:\mathcal{N}\rightarrow\mathcal{M}\) be smooth, and \(c\) is a regular value and \({f}^{-1}\left(c\right)\) is nonempty, then \({F}^{-1}\left(c\right)\) is a regular submanifold of \(\mathcal{N}\) of dimension \(n-m\).
| Examples: The general linear group \(\text{GL}\left(n,\mathbb{R}\right)\) is a \({n}^{2}\) dimensional regular submanifold of \({\mathbb{R}}^{ {n}^{2}}\). The special linear group \(\text{SL}\left(n,\mathbb{R}\right)\) is a \({n}^{2}-1\) dimensional regular submanifold of general linear group \(\text{GL}\left(n,\mathbb{R}\right)\), (let $$f\left(A\right)=\left | A\right | $$). |
Related concepts: maximal rank, constant rank theorem, immersion theorem, submersion theorem, constant rank level set theorem, embedding
If \(F:\mathcal{N}\rightarrow\mathcal{M}\) is smooth and has maximal rank at \(p\), then there are three possibilites:
(Constant Rank) Let \(f:\mathcal{N}\rightarrow\mathcal{M}\) be smooth and has constant rank \(k\) in a neighbourhood of \(p\in\mathcal{N}\), then there exist charts \(\left(U,\phi\right),\left(V,\psi\right)\) centered at \(p\), \(f\left(p\right)\) such that
\[\psi\circ f\circ{\phi}^{-1}\left({r}_{1},{r}_{2},\dots,{r}_{n}\right)=\left({r}_{1},{r}_{2},\dots,{r}_{k},0,\dots,0\right).\]Having maximal rank is an open condition, i.e. if \(f\) has maximal rank at \(p\), then there exists a open neighborhood \(U\) of \(p\) such that \(f\) has maximal rank in \(U\).
(Immersion) Let \(f:\mathcal{N}\rightarrow\mathcal{M}\) be smooth and is an immersion at \(p\in\mathcal{N}\), then there exist charts \(\left(U,\phi\right),\left(V,\psi\right)\) centered at \(p\), \(f\left(p\right)\) such that
\[\psi\circ f\circ{\phi}^{-1}\left({r}_{1},{r}_{2},\dots,{r}_{n}\right)=\left({r}_{1},{r}_{2},\dots,{r}_{n},0,\dots,0\right).\]
(Submersion) Let \(f:\mathcal{N}\rightarrow\mathcal{M}\) be smooth and is a submersion at \(p\in\mathcal{N}\), then there exist charts \(\left(U,\phi\right),\left(V,\psi\right)\) centered at \(p\), \(f\left(p\right)\) such that
\[\psi\circ f\circ{\phi}^{-1}\left({r}_{1},{r}_{2},\dots,{r}_{n}\right)=\left({r}_{1},{r}_{2},\dots,{r}_{m}\right).\]
(Constant Rank Level Set) Let \(F:\mathcal{N}\rightarrow\mathcal{M}\) be smooth and has constant rank \(k\) in a neighbourhood of \({f}^{-1}\left(c\right)\), then \({f}^{-1}\left(c\right)\) is a regular submanifold of dimension \(n-k\).
The regular level set theorem is a specilaized version of constant rank level set thoerem.
An example: The orthogonal group \(O\left(n\right)\) is a regular submanifold of \(\text{GL}\left(n\right)\), (let \(f\left(A\right)={A}^{T}A-I\)).
A smooth map \(f:\mathcal{N}\rightarrow\mathcal{M}\) is an embedding iff \(f\) is an one to one immersion and the topology of \(f\left(\mathcal{N}\right)\) induced by \(f\) is homeomorphism to subspace topology.
If \(f\) is an embedding, \(f\left(\mathcal{N}\right)\) is an regular submanifold.
Related concepts: tangent bundle, natural map, vector field, induced map
Tangent bundle \(T\mathcal{M}\) of manifold \(\mathcal{M}\) is
\[T\mathcal{M}=\bigcup_{p\in\mathcal{M}}^{}{T}_{p}\mathcal{M}.\]
The natural map is \(\pi:T\mathcal{M}\rightarrow\mathcal{M}\)
\[\pi\left(v\right)=p,\quad\text{if }v\in{T}_{p}\mathcal{M}.\]
the tangent bundle \(T\mathcal{M}\) has a manifold structure.
For each chart \(\left(U,\phi\right)\) of \(\mathcal{M}\) centered at \(p\in U\), the tangent space \({T}_{p}U\) equals \({T}_{p}\mathcal{M}\), we denotes the union of tangent spaces of \(U\) as \(T U=\bigcup_{p\in U}^{}{T}_{p}\mathcal{M}\). Let \(\nu\in T U\), we set \(\widetilde{\phi}:T U\rightarrow\phi\left(U\right)\times{\mathbb{R}}^{n}\)
\[\widetilde{\phi}\left(\nu\right)=\left(\phi\circ\pi\left(\nu\right),c\left(\nu\right)\right)\]where \(c\left(\nu\right)\) satisfies
\[\nu={c}^{1}\left(\nu\right)\frac{\partial}{\partial{x}^{1}}+{c}^{2}\left(\nu\right)\frac{\partial}{\partial{x}^{2}}+\dots+{c}^{n}\left(\nu\right)\frac{\partial}{\partial{x}^{n}}\]This is a diffeomorphism to \(\phi\left(U\right)\times{\mathbb{R}}^{n}\), and we give \(T U\) the topology induced by \(\widetilde{\phi}\). The topology of \(T\mathcal{M}\) is generated from the topology of all charts \(\left(U,\phi\right)\).
A vector field \(X\) on \(\mathcal{M}\) is a function assigns a tangent vector \({X}_{p}\in T\mathcal{M}\) to each point \(p\) of \(\mathcal{M}\).
Let \(X\) be a smooth vector field on \(\mathcal{M}\), \(f\in{C}^{\infty}\left(\mathcal{M}\right)\), the induced map of \(X\) is \(X f:\mathcal{M}\rightarrow\mathbb{R}\)
\[{\left(X f\right)}_{p}={X}_{p}f.\]\(X\) can be seen as a map from \({C}^{\infty}\left(\mathcal{M}\right)\) to \({C}^{\infty}\left(\mathcal{M}\right)\), then \(X\) is a derivative.