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 ${\mathcal {C}}^{n}(M)$ functions.

Definition of a submanifold

Definition 4.1:

Let $M$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$, and let $A$ be it's maximal atlas. If $m\in \{1,\ldots ,d\}$, we call a subset $N\subseteq M$ a submanifold of dimension $m$ iff $m$ is the largest number in $\{1,\ldots ,d\}$ such that for each $p\in N$ there exists $(O,\phi )\in A$ such that $p\in O$ and

$\phi (O\cap N)\subseteq \{(x_{1},\ldots ,x_{d})\in \mathbb {R} ^{d}|x_{d-m+1}=x_{d-m+2}=\ldots =x_{d}=0\}$

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

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

Proof:

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

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

2. We show that $\phi |_{V}$ is compatible with all $(U,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$.

Let $(U,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$.

We have:

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

and

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

, 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 $\phi$ are compatible. Due to the definitions of ${\mathcal {C}}$ and ${\mathcal {C}}$ respectively, the lemma is proved.$\Box$

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

Proof:

1. We show that $\Phi \circ \phi$ is a chart.

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

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

Let $(U,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$.

We have:

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

And also:

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

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 $\phi$ and $\theta$ are compatible. By definition of ${\mathcal {C}}^{n}((\Phi \circ \phi )(O),\mathbb {R} ^{d})$ and ${\mathcal {C}}^{n}(\psi (O),\mathbb {R} ^{d})$ respectively, we are finished with the proof of this lemma.$\Box$

Theorem 4.4:

Let $M$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$, where $n$ must be $\geq 1$ here, with maximal atlas $A$, let $m\in \{1,\ldots ,d\}$ and let $f_{1},\ldots ,f_{m}\in {\mathcal {C}}^{n}(M)$ (remember def. 1.5). If for each $p\in N$ there exists $(O,\phi )\in A$ such that $p\in O$ and the matrix

${\begin{pmatrix}\left(\partial _{x_{1}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{d}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))\\\vdots &\ddots &\vdots \\\left(\partial _{x_{1}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{d}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))\end{pmatrix}}$

has rank $m$, then the set $\left\{q\in M|f_{1}(q)=\ldots =f_{m}(q)=0\right\}$ is a submanifold of dimension $d-m$ of $M$.

Proof:

Since the matrix

${\begin{pmatrix}\left(\partial _{x_{1}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{d}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))\\\vdots &\ddots &\vdots \\\left(\partial _{x_{1}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{d}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))\end{pmatrix}}$

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

${\begin{pmatrix}\left(\partial _{x_{\sigma (1)}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{\sigma (d)}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))\\\vdots &\ddots &\vdots \\\left(\partial _{x_{\sigma (1)}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{\sigma (d)}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))\end{pmatrix}}$

Hence, the $d\times d$ matrix

${\begin{pmatrix}1&0&\cdots &0&0&0&\cdots &0\\0&1&\cdots &0&0&&&\\\vdots &\ddots &\ddots &\ddots &\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1&0&&&\\0&0&\cdots &0&1&0&\cdots &0\\\left(\partial _{x_{\sigma (1)}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))&&\cdots &&&&\cdots &\left(\partial _{x_{\sigma (d)}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))\\\vdots &&&&\ddots &&&\vdots \\\left(\partial _{x_{\sigma (1)}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))&&\cdots &&&&\cdots &\left(\partial _{x_{\sigma (d)}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))\end{pmatrix}}$

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 $\Phi :\mathbb {R} ^{d}\to \mathbb {R} ^{d}$ given by

$\Phi (x_{1},\ldots ,x_{d})={\begin{pmatrix}x_{1}\\\vdots \\x_{d-m}\\(f_{1}\circ \phi ^{-1})(x_{\sigma (1)},\ldots ,x_{\sigma (d)}\\\vdots \\(f_{m}\circ \phi ^{-1})(x_{\sigma (1)},\ldots ,x_{\sigma (d)})\end{pmatrix}}$

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

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

$\Phi \circ \phi |_{\phi ^{-1}(V)}(q)=(\phi _{1}(q),\ldots ,\phi _{d-m}(q),f_{1}(q),\ldots ,f_{m}(q))=(\phi _{1}(q),\ldots ,\phi _{d-m}(q),0,\ldots ,0)$

Hence, $\left\{q\in M|f_{1}(q)=\ldots =f_{m}(q)=0\right\}$ is a submanifold of dimension $d-m$.$\Box$

Sources

• Torres del Castillo, Gerardo (2012). Differentiable Manifolds. Boston: Birkhäuser. ISBN 978-0-8176-8271-2.