Differentiable Manifolds/Submanifolds

In this chapter, we will show what submanifolds are, and how we can obtain, under a condition, a submanifold out of some ${𝒞}^n(M)$ functions.

How to obtain a submanifold out of a set of certain functions

Lemma 4.2: Let $M$ be a $d$-dimensional manifold of class ${𝒞}^n$ with atlas $\{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$, let $A$ be it's maximal atlas, let $(O,\varphi )\in A$, and let $V\subseteq O$ be an open subset of $O$. Then $(V,\varphi {|}_V)\in A$.

Proof:

1. We show that $\varphi {|}_V:V\to \varphi (V)$ is a chart.

It is a homeomorphism since the restriction of a homeomorphism is a homeomorphism, and if $V\subseteq O$ is open, then $\varphi (V)$ is open in $\varphi (O)$ since $\varphi$ is a homeomorphism, and further, due to the definition of the subspace topology and since $\varphi (V)$ is open in $\varphi (O)$, we have $\varphi (V)=W\cap \varphi (O)$ for an open set $W\subseteq {ℝ}^d$, and hence $\varphi (V)$ is open in ${ℝ}^d$ as the intersection of two open sets.

2. We show that $\varphi {|}_V$ is compatible with all $(U,\theta )\in \{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$.

Let $(U,\theta )\in \{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$.

We have:

$\theta {|}_{V\cap U}\circ (\varphi {|}_V){|}_{V\cap U}^{-1}=(\theta {|}_{O\cap U}\circ \varphi {|}_{O\cap U}^{-1}){|}_{\varphi (V\cap U)}$

and

$(\varphi {|}_V){|}_{V\cap U}\circ {\theta }^{-1}{|}_{V\cap U}=(\varphi {|}_{O\cap U}\circ \theta {|}_{O\cap U}^{-1}){|}_{\theta (V\cap U)}$

, which can be verified by direct calculation. But these are $n$-times differentiable (or continuous if $n=0$), since they are restrictions of $n$-times differentiable (or continuous if $n=0$) functions; this is since $\theta$ and $\varphi$ are compatible. Due to the definitions of $𝒞$ and $𝒞$ respectively, the lemma is proved.$◻$

Lemma 4.3: Let $M$ be a $d$-dimensional manifold of class ${𝒞}^n$ with atlas $\{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$, let $A$ be it's maximal atlas, let $(O,\varphi )\in A$, and let $\mathrm{\Phi }:\varphi (O)\to {ℝ}^d$ be a diffeomorphism of class ${𝒞}^n$. Then we have: $(O,\mathrm{\Phi }\circ \varphi )\in A$.

Proof:

1. We show that $\mathrm{\Phi }\circ \varphi$ is a chart.

By invariance of domain, and since $\varphi (O)$ is open in ${ℝ}^d$ since $\varphi$ is a chart, $(\mathrm{\Phi }\circ \varphi )(O)$ is open in ${ℝ}^d$. Furthermore, $\mathrm{\Phi }$ and $\varphi$ are homeomorphisms ($\mathrm{\Phi }$ is a homeomorphism because every diffeomorphism is a homeomorphism), and therefore $\mathrm{\Phi }\circ \varphi$ is a homeomorphism as well. Thus, $\mathrm{\Phi }\circ \varphi$ is a chart.

2. We show that $\mathrm{\Phi }\circ \varphi$ is compatible with all $(U,\theta )\in \{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$.

Let $(U,\theta )\in \{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$.

We have:

$\theta {|}_{U\cap O}\circ (\mathrm{\Phi }\circ \varphi ){|}_{U\cap O}^{-1}=\theta {|}_{U\cap O}\circ \varphi {|}_{U\cap O}^{-1}\circ \mathrm{\Phi }{|}_{\varphi (O\cap U)}^{-1}$

And also:

$(\mathrm{\Phi }\circ \varphi ){|}_{U\cap O}\circ \theta {|}_{U\cap O}^{-1}=\mathrm{\Phi }{|}_{\varphi (O\cap U)}\circ \varphi {|}_{O\cup U}\circ \theta {|}_{U\cap O}^{-1}$

These functions are $n$-times differentiable (or continuous if $n=0$), because they are compositions of functions, which are $n$-times differentiable (or continuous if $n=0$); this is since $\varphi$ and $\theta$ are compatible. By definition of ${𝒞}^n((\mathrm{\Phi }\circ \varphi )(O),{ℝ}^d)$ and ${𝒞}^n(\psi (O),{ℝ}^d)$ respectively, we are finished with the proof of this lemma.$◻$

Proof:

Since the matrix

$\left(\begin{array}{ccc}({\partial }_{x_1}(f_1\circ {\varphi }^{-1}))(\varphi (p))& \cdots & ({\partial }_{x_d}(f_1\circ {\varphi }^{-1}))(\varphi (p))\\ ⋮& \ddots & ⋮\\ ({\partial }_{x_1}(f_m\circ {\varphi }^{-1}))(\varphi (p))& \cdots & ({\partial }_{x_d}(f_m\circ {\varphi }^{-1}))(\varphi (p))\end{array}\right)$

has rank $m$, it has $m$ linearly independent columns (this is a theorem from linear algebra). Therefore there exists a permutation $\sigma :\{1,\dots ,d\}\to \{1,\dots ,d\}$ such that the last $m$ columns of thee matrix

$\left(\begin{array}{ccc}({\partial }_{x_{\sigma (1)}}(f_1\circ {\varphi }^{-1}))(\varphi (p))& \cdots & ({\partial }_{x_{\sigma (d)}}(f_1\circ {\varphi }^{-1}))(\varphi (p))\\ ⋮& \ddots & ⋮\\ ({\partial }_{x_{\sigma (1)}}(f_m\circ {\varphi }^{-1}))(\varphi (p))& \cdots & ({\partial }_{x_{\sigma (d)}}(f_m\circ {\varphi }^{-1}))(\varphi (p))\end{array}\right)$

Hence, the $d\times d$ matrix

$\left(\begin{array}{cccccccc}1& 0& \cdots & 0& 0& 0& \cdots & 0\\ 0& 1& \cdots & 0& 0& & & \\ ⋮& \ddots & \ddots & \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 1& 0& & & \\ 0& 0& \cdots & 0& 1& 0& \cdots & 0\\ ({\partial }_{x_{\sigma (1)}}(f_1\circ {\varphi }^{-1}))(\varphi (p))& & \cdots & & & & \cdots & ({\partial }_{x_{\sigma (d)}}(f_1\circ {\varphi }^{-1}))(\varphi (p))\\ ⋮& & & & \ddots & & & ⋮\\ ({\partial }_{x_{\sigma (1)}}(f_m\circ {\varphi }^{-1}))(\varphi (p))& & \cdots & & & & \cdots & ({\partial }_{x_{\sigma (d)}}(f_m\circ {\varphi }^{-1}))(\varphi (p))\end{array}\right)$

is invertible (one can prove the invertibility of the transpose by induction and Laplace's formula). But the latter matrix is the Jacobian matrix of the function $\mathrm{\Phi }:{ℝ}^d\to {ℝ}^d$ given by

$\mathrm{\Phi }(x_1,\dots ,x_d)=\left(\begin{array}{c}{x}_1\\ ⋮\\ x_{d-m}\\ (f_1\circ {\varphi }^{-1})(x_{\sigma (1)},\dots ,x_{\sigma (d)}\\ ⋮\\ (f_m\circ {\varphi }^{-1})(x_{\sigma (1)},\dots ,x_{\sigma (d)})\end{array}\right)$

at $\varphi (p)$. By the inverse function theorem, there exists an open set $V\subseteq {ℝ}^d$ such that $\varphi (p)\in V$ and $\mathrm{\Phi }{|}_V$ is a diffeomorphism.

Since $\varphi$ is a homeomorphism, and in particular is continuous, ${\varphi }^{-1}(V)$ is an open subset of $O$. Due to lemma 4.2, $({\varphi }^{-1}(V),\varphi {|}_{{\varphi }^{-1}(V)})\in A$. Due to lemma 4.3, $({\varphi }^{-1}(V),\mathrm{\Phi }\circ \varphi {|}_{{\varphi }^{-1}(V)})\in A$. But it also holds that for $q$ such that $f_1(q)=\dots =f_m(q)=0$:

$\mathrm{\Phi }\circ \varphi {|}_{{\varphi }^{-1}(V)}(q)=({\varphi }_1(q),\dots ,{\varphi }_{d-m}(q),f_1(q),\dots ,f_m(q))=({\varphi }_1(q),\dots ,{\varphi }_{d-m}(q),0,\dots ,0)$

Hence, $\{q\in M|f_1(q)=\dots =f_m(q)=0\}$ is a submanifold of dimension $d-m$.$◻$

Sources

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