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.

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.

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.

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 $$d - m$$.