Differentiable Manifolds/Maximal atlases, second-countable spaces and partitions of unity

Maximal atlases
{{TextBox| M=0 | W=100% | BG=#FFFFFF |1=Definition 3.1:

Let $$M$$ be a $$d$$-dimensional manifold of class $$\mathcal C^n$$ and let $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon \}$$ be it's atlas. We call the set
 * $$A := \left\{(U, \theta) | U \subseteq M \text{ open }, \theta: U \to \mathbb R^d, \theta \text{ is compatible of class } \mathcal C^n \text{ to all } \phi_\upsilon, \upsilon \in \Upsilon\} \right\}$$

the maximal atlas of $$M$$.}}

Lemma 3.2: We have $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon \} \subseteq A$$.

Proof: This is because if $$(O, \phi) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon \}$$, then by definition of an atlas it is compatible with all the elements of $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon \}$$ and hence, by definition of $$A$$, contained in $$A$$.

Theorem 3.3: The maximal atlas really is an atlas; i. e. for every point $$p \in M$$ there exists $$(U, \theta)$$ such that $$p \in M$$, and every two charts in it are compatible.

Proof:

1.

We first show that for every point $$p \in M$$ there exists $$(U, \theta)$$ such that $$p \in M$$:

From lemma 3.2 we know that the atlas of $$M$$ is contained in $$A$$.

Let now $$p \in M$$. Due to the definition of an atlas, we find an $$(U, \theta) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon \}$$ such that $$p \in O$$. Since $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon \} \subseteq A$$, we obtain $$(U, \theta) \in A$$.

2.

We prove that every two charts $$\phi: O \to \mathbb R^d, \theta: U \to \mathbb R^d$$ such that $$(O, \phi), (U, \theta) \in A$$, are compatible.

So let $$\phi: O \to \mathbb R^d, \theta: U \to \mathbb R^d$$ such that $$(O, \phi), (U, \theta) \in A$$ be 'arbitrary' (of course we still require $$(O, \phi), (U, \theta) \in A$$).

If we have $$O \cap U = \emptyset$$, this directly implies compatibility (recall that we defined compatibility so that if $$U \cap O = \emptyset$$ for two charts $$\phi: O \to \mathbb R^d, \theta: U \to \mathbb R^d$$, then the two are by definition automatically compatible).

So in this case, we are finished. Now we shall prove the other case, which namely is $$O \cap U \neq \emptyset$$.

Due to the definition of compatibility of class $$\mathcal C^n$$, we have to prove that the function


 * $$\phi|_{U \cap O} \circ \psi|_{U \cap O}^{-1} : \psi(U \cap O) \to \phi(U \cap O)$$

is contained in $$\mathcal C^n(\psi(U \cap O), \mathbb R^d)$$ and


 * $$\psi|_{U \cap O} \circ \phi|_{U \cap O}^{-1} : \phi(U \cap O) \to \psi(U \cap O)$$

is contained in $$\mathcal C^n(\phi(U \cap O), \mathbb R^d)$$.

Let $$p \in O \cap U$$. Since $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon \}$$ is the atlas of $$M$$, we find a chart $$(\chi, V) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon \}$$ such that $$p \in V$$. Due to the definition of $$A$$, $$\chi$$ and $$\psi$$ are compatible and $$\chi$$ and $$\phi$$ are compatible. Hence, the functions
 * $$\phi|_{V \cap U \cap O} \circ \chi|_{V \cap U \cap O}^{-1} \circ \chi|_{V \cap U \cap O} \circ \psi|_{V \cap U \cap O}^{-1} : \psi(V \cap U \cap O) \to \phi(V \cap U \cap O)$$

and
 * $$\psi|_{V \cap U \cap O} \circ \chi|_{V \cap U \cap O}^{-1} \circ \chi|_{V \cap U \cap O} \circ \phi|_{V \cap U \cap O}^{-1} : \psi(V \cap U \cap O) \to \phi(V \cap U \cap O)$$

are $$n$$-times differentiable (or, if $$n=0$$, continuous), in particular at $$\psi(p)$$, $$\phi(p)$$ respectively. Since $$p \in O \cap U$$ was arbitrary, since
 * $$\phi|_{V \cap U \cap O} \circ \chi|_{V \cap U \cap O}^{-1} \circ \chi|_{V \cap U \cap O} \circ \psi|_{V \cap U \cap O}^{-1} = (\phi|_{U \cap O} \circ \psi|_{U \cap O}^{-1})|_{\psi(V \cap U \cap O)}$$

and
 * $$\psi|_{V \cap U \cap O} \circ \chi|_{V \cap U \cap O}^{-1} \circ \chi|_{V \cap U \cap O} \circ \phi|_{V \cap U \cap O}^{-1} = (\psi|_{U \cap O} \circ \phi|_{U \cap O}^{-1})|_{\phi(V \cap U \cap O)}$$

(which you can show by direct calculation!) and since $$\phi, \psi$$ are bijective, this shows the theorem.

This is, in fact, the reason why the word maximal atlas for $$A$$ does not completely miss the point.

Proof: We show that there does not exist an atlas $$B$$ of $$M$$ such that $$A \subsetneq B$$.

Assume by contradiction that there exists such an atlas. Then we find an element $$(U, \theta) \in B \setminus A$$. But since $$B$$ is an atlas, $$\theta$$ is compatible to all other charts $$\phi$$ for which $$(O, \phi) \in A$$. This means, due to lemma 3.2, that it is compatible to every $$\phi_\upsilon, \upsilon \in \Upsilon$$. Hence, due to the definition of $$A$$, $$(U, \theta) \in A$$. This is a contradiction!

Locally finite refinements and partitions of unity
Example 3.8:

The set $$\{B_\epsilon(x) | x \in \mathbb R^d, \epsilon > 0 \}$$ is an open cover of the real numbers.

We will now prove a few lemmas, which will help us to prove that every manifold whose topology has a countable basis admits partition of unity. Then, we will prove that every manifold whose topology has a countable basis admits partition of unity :-)

Lemma 3.11:

Let $$M$$ be a manifold with a countable basis. Then $$M$$ has a countable basis $$\{V_j | j \in \mathbb N \}$$ such that for each $$j \in \mathbb N$$, $$\overline{U_j}$$ is compact.

Proof:

Let $$\{U_k | k \in \mathbb N\}$$ be a countable basis of $$M$$. For each $$p \in M$$, we choose a chart $$(O, \phi)$$ such that $$p \in O$$. Then we choose $$W_p := \phi(O) \cap B_1(\phi(p))$$. Since in $$\mathbb R^d$$, sets are compact if and only if bounded and closed, $$\overline{W_p}$$ is compact. There is a theorem from topology, which states that the image of a compact set under a homeomorphism is again compact. Hence, $$\phi^{-1}(\overline{W_p})$$ is a compact subset of $$O$$.

Further, if $$\{V_\kappa, \kappa \in \Kappa \}$$ is an cover of $$\phi^{-1}(\overline{W_p})$$ by open subsets of $$M$$, then the set $$\{V_\kappa \cap O, \kappa \in \Kappa \}$$ is a cover of $$\phi^{-1}(\overline{W_p})$$ by open subsets of $$O$$. Since $$\phi^{-1}(\overline{W_p})$$ is compact in $$O$$, we may pick out of the latter a finite subcover $$V_{\kappa_1} \cap O, \ldots, V_{\kappa_n} \cap O$$. Then, since
 * $$O \subseteq \bigcup_{j=1}^n V_{\kappa_j} \cap O \subseteq \subseteq \bigcup_{j=1}^n V_{\kappa_j}$$

, the set $$V_{\kappa_1}, \ldots, V_{\kappa_n}$$ is a finite subcover of $$\{V_\kappa, \kappa \in \Kappa \}$$. Thus, $$\phi^{-1}(\overline{W_p})$$ is also a compact subset of $$M$$.

As $$\phi$$ is a homeomorphism, $$W_p$$ is open in $$M$$, and from $$W_p \subset \overline{W_p}$$, it follows $$\phi^{-1}(W_p) \subset \phi^{-1}(\overline{W_p})$$. Thus, also
 * $$\overline{\phi^{-1}(W_p)} \subseteq \phi^{-1}(\overline{W_p})$$

since the closure of $$\phi^{-1}(W_p)$$ is, by definition (with the definition of some lectures), equal to
 * $$\bigcap_{A \supseteq \phi^{-1}(W_p) \atop A \text{ closed }} A$$

Further, another theorem from topology states that closed subsets of compact sets are compact. Hence, $$\overline{\phi^{-1}(W_p)}$$ is compact.

Since $$\{U_k | k \in \mathbb N\}$$ was a basis, each of the $$\phi^{-1}(W_p)$$ can be written as the union of elements of $$\{U_k | k \in \mathbb N\}$$. We choose now our new basis as consisting of the union over $$p \in M$$ of the elements of $$\{U_k | k \in \mathbb N\}$$ with smallest index $$m_p$$, such that $$p \in U_{m_p}$$ and $$U_{m_p} \subseteq \phi^{-1}(W_p)$$. Now the closures of the $$U_{m_p}$$ are compact: From $$U_{m_p} \subseteq \phi^{-1}(W_p)$$ follows that $$\overline{U_{m_p}} \subseteq \overline{\phi^{-1}(W_p)}$$, and since $$\overline{\phi^{-1}(W_p)} \subseteq \phi^{-1}(\overline{W_p})$$, $$\overline{U_{m_p}}$$ is compact as the closed subset of a compact set.

Since our new basis is a subset of a countable set, it is itself countable (we include finite sets in the category 'countable' here). Thus, we have obtained a countable basis the elements of which have compact closure.

Lemma 3.12:

Let $$M$$ be a manifold with a countable base (i. e. a second-countable manifold). Then for every cover of $$M$$ there is a locally finite refinement.

Proof:

Let $$\{V_\kappa | \kappa \in \Kappa\}$$ be a cover of $$M$$. Due to lemma 3.11, we may choose a countable basis $$\{U_j | j \in \mathbb N \}$$ of $$M$$ such that each $$\overline{U_j}$$ is compact. We now define a sequence of compact sets $$(A_l)_{l \in \mathbb N}$$ inductively as follows: We set $$A_1 := \overline{U_1}$$. Once we defined $$A_l$$, we define
 * $$A_{l+1} := \overline{U_1} \cup \cdots \cup \overline{U_m}$$

, where $$m \in \mathbb N$$ is smallest such that we have:
 * $$\text{int } A_l \subset \overline{U_1} \cup \cdots \cup \overline{U_m}$$

This is compact, since a theorem from topology states that the finite union of compact sets is compact. Since, as mentioned before, there is a theorem from topology stating that closed subsets of compact sets are compact, the sets defined by $$B_1 := A_1$$ and
 * $$B_l := A_l \setminus \text{int } A_{l-1}$$

for $$l \ge 2$$ (intuitively the closed annulus) are compact. Further, the sets $$C_l$$, defined by $$C_1 := \text{int } A_2$$, $$C_2 := \text{int } A_3$$ and
 * $$C_l := \text{int } A_{l+1} \setminus A_{l-2}$$

for $$l \ge 3$$ (intuitively the next bigger open annulus) are open, and we have for all $$l \in \mathbb N$$:
 * $$B_l \subset C_l$$

Now since $$M$$ is covered by $$\{V_\kappa | \kappa \in \Kappa\}$$, so is each of the sets $$B_l$$. Now we compose our locally finite refinement as follows: We include all the sets, which are the intersection of $$C_l$$ and the (by compactness existing) sets of the finite subcovers of $$B_l$$ out of $$\{V_\kappa | \kappa \in \Kappa\}$$. This is a locally finite refinement.

Lemma 3.13:

Let $$M$$ be a $$d$$-dimensional manifold of class $$\mathcal C^n$$ with atlas $$\{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, let $$(O, \phi) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$, let $$W \subseteq O$$ be open in $$O$$ (with respect to the subspace topology and let $$p \in O$$ and let $$\epsilon > 0$$ be such that $$B_\epsilon(\phi(p)) \subseteq \phi(O \cap W)$$. If we define
 * $$\eta: \mathbb R^d \to \mathbb R, \eta(x) = \begin{cases}

e^{1-\frac{1}{1-\|x\|^2}}& \text{ if } \|x\|_2 < 1\\ 0 & \text{ if } \|x\|_2 \geq 1 \end{cases}$$ , and
 * $$h_{p, W, \phi}: M \to \mathbb R, h_{p, W, \phi}(q) := \begin{cases}

\eta\left( \frac{1}{\epsilon} (\phi(p) - \phi(q)) \right) & q \in O \\ 0 & q \notin O \\ \end{cases}$$, then we have $$h_{p, W, \phi} \in \mathcal C^n(M)$$.

Proof:

Let $$(U, \theta) \in \{(O_\upsilon, \phi_\upsilon) | \upsilon \in \Upsilon\}$$. Then we have for $$x \in \theta(U)$$:


 * $$(h_{p, W, \phi}|_U \circ \theta^{-1})(x) = \begin{cases}

0 & (\phi \circ \theta^{-1})(x) \notin B_\epsilon(\phi(p)) \\ e^{1-\frac{1}{1-\|\frac{1}{\epsilon} \left( \phi(p) - (\phi \circ \theta^{-1})(x) \right)\|^2}} & (\phi \circ \theta^{-1})(x) \in B_\epsilon(\phi(p)) \end{cases}$$ This function is $$n$$ times differentiable (or continuous if $$n=0$$) as the composition of $$n$$ times differentiable (or continuous if $$n=0$$) functions.

Proof:

Let $$\{V_\kappa | \kappa \in \Kappa\}$$ be an open cover of $$M$$.

We choose for each point $$p \in M$$ an atlas $$(O, \phi)$$ such that $$p \in O$$. Further, we choose an arbitrary $$V_{\kappa_p}$$ in the open cover such that $$p \in V_{\kappa_p}$$. By definition of the subspace topology we have that $$V_{\kappa_p} \cap O$$ is open in $$O$$. Therefore, due to lemma 3.13, we may choose $$h_{p, V_{\kappa_p} \cap O, \phi} \in \mathcal C^n(M)$$ such that $$p \in \{q \in M | h_{p, V_{\kappa_p} \cap O, \phi}(q) > 0\} =: W_p$$. Since $$h_{p, V_{\kappa_p} \cap O, \phi}$$ is continuous, all the $$W_p$$ are open; this is because they are preimages of the open set $$(0, \infty)$$. Further, since there is a $$W_p$$ for every $$p \in M$$, and always $$p \in W_p$$, the $$W_p$$ form a cover of $$M$$. Due to lemma 3.12 we may choose a locally finite refinement. This open cover, this set of open sets we shall denote by $$S$$.

We now define the function
 * $$\varphi: M \to \mathbb R, \varphi(q) := \sum_{W_p \in S} h_{p, V_{\kappa_p} \cap O, \phi}(q)$$

This function is of class $$\mathcal C^n(M)$$ as a finite sum (because for each $$p$$ there are only finitely many $$W \in S$$ such that $$p \in W$$, because $$S$$ was a locally finite subcover) of $$\mathcal C^n(M)$$ functions (that finite sums of $$\mathcal C^n(M)$$ functions are again $$\mathcal C^n(M)$$ follows from theorem 2.22 and induction) and does not vanish anywhere (since for every $$p$$ there is a $$W_q \in S$$ such that $$p$$ is in it; remember that a finite refinement is an open cover), and therefore follows from theorem 2.26, that all the functions $$\varphi_{p, V_{\kappa_p} \cap O, \phi} := \frac{h_{p, V_{\kappa_p} \cap O, \phi}}{\varphi}$$ are contained in $$\mathcal C^n(M)$$. It is not difficult to show that these functions are non-negative and that they sum up to $$1$$ at every point. Further due to the construction, each of their supports is contained in one $$V_\kappa$$. Thus they form the desired partition of unity.