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

Maximal atlases

Lemma 3.2: We have $\{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}\subseteq A$.

Proof: This is because if $(O,\varphi )\in \{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$, then by definition of an atlas it is compatible with all the elements of $\{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$ 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 $(U,\theta )\in \{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$ such that $p\in O$. Since $\{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}\subseteq A$, we obtain $(U,\theta )\in A$.

2.

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

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

If we have $O\cap U=\mathrm{\varnothing }$, this directly implies compatibility (recall that we defined compatibility so that if $U\cap O=\mathrm{\varnothing }$ for two charts $\varphi :O\to {ℝ}^d,\theta :U\to {ℝ}^d$, 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 $O\cap U\ne \mathrm{\varnothing }$.

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

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

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

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

is contained in ${𝒞}^n(\varphi (U\cap O),{ℝ}^d)$.

Let $p\in O\cap U$. Since $\{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$ is the atlas of $M$, we find a chart $(\chi ,V)\in \{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$ such that $p\in V$. Due to the definition of $A$, $\chi$ and $\psi$ are compatible and $\chi$ and $\varphi$ are compatible. Hence, the functions

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

and

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

are $n$-times differentiable (or, if $n=0$, continuous), in particular at $\psi (p)$, $\varphi (p)$ respectively. Since $p\in O\cap U$ was arbitrary, since

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

and

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

(which you can show by direct calculation!) and since $\varphi ,\psi$ 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 $A⊊B$.

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 $\varphi$ for which $(O,\varphi )\in A$. This means, due to lemma 3.2, that it is compatible to every ${\varphi }_{\upsilon },\upsilon \in \mathrm{{\rm Y}}$. 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 $\{B_{ϵ}(x)|x\in {ℝ}^d,ϵ>0\}$ 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 $(O,\varphi )$ such that $p\in O$. Then we choose $W_p:=\varphi (O)\cap B_1(\varphi (p))$. Since in ${ℝ}^d$, 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, ${\varphi }^{-1}(\overline{W_p})$ is a compact subset of $O$.

Further, if $\{V_{\kappa },\kappa \in \mathrm{K}\}$ is an cover of ${\varphi }^{-1}(\overline{W_p})$ by open subsets of $M$, then the set $\{V_{\kappa }\cap O,\kappa \in \mathrm{K}\}$ is a cover of ${\varphi }^{-1}(\overline{W_p})$ by open subsets of $O$. Since ${\varphi }^{-1}(\overline{W_p})$ is compact in $O$, we may pick out of the latter a finite subcover $V_{{\kappa }_1}\cap O,\dots ,V_{{\kappa }_n}\cap O$. Then, since

$O\subseteq \bigcup _{j=1}^n{V}_{{\kappa }_j}\cap O\subseteq \subseteq \bigcup _{j=1}^n{V}_{{\kappa }_j}$

, the set $V_{{\kappa }_1},\dots ,V_{{\kappa }_n}$ is a finite subcover of $\{V_{\kappa },\kappa \in \mathrm{K}\}$. Thus, ${\varphi }^{-1}(\overline{W_p})$ is also a compact subset of $M$.

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

$\overline{{\varphi }^{-1}(W_p)}\subseteq {\varphi }^{-1}(\overline{W_p})$

since the closure of ${\varphi }^{-1}(W_p)$ is, by definition (with the definition of some lectures), equal to

$\bigcap _{\genfrac{}{}{0ex}{}{A\supseteq {\varphi }^{-1}(W_p)}{A\text{ closed }}}A$

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

Since $\{U_k|k\in \mathbb{N}\}$ was a basis, each of the ${\varphi }^{-1}(W_p)$ 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 $U_{m_p}\subseteq {\varphi }^{-1}(W_p)$. Now the closures of the $U_{m_p}$ are compact: From $U_{m_p}\subseteq {\varphi }^{-1}(W_p)$ follows that $\overline{U_{m_p}}\subseteq \overline{{\varphi }^{-1}(W_p)}$, and since $\overline{{\varphi }^{-1}(W_p)}\subseteq {\varphi }^{-1}(\overline{W_p})$, $\overline{U_{m_p}}$ 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 $\{V_{\kappa }|\kappa \in \mathrm{K}\}$ 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 $\{V_{\kappa }|\kappa \in \mathrm{K}\}$, 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 $\{V_{\kappa }|\kappa \in \mathrm{K}\}$. This is a locally finite refinement.$◻$

Lemma 3.13:

Let $M$ be a $d$-dimensional manifold of class ${𝒞}^n$ with atlas $\{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$, let $(O,\varphi )\in \{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$, let $W\subseteq O$ be open in $O$ (with respect to the subspace topology and let $p\in O$ and let $ϵ>0$ be such that $B_{ϵ}(\varphi (p))\subseteq \varphi (O\cap W)$. If we define

$\eta :{ℝ}^d\to ℝ,\eta (x)=\{\begin{array}{cc}{e}^{1-\frac{1}{1-\Vert x{\Vert }^2}}& \text{ if }\Vert x{\Vert }_2<1\\ 0& \text{ if }\Vert x{\Vert }_2\ge 1\end{array}$

, and

$h_{p,W,\varphi }:M\to ℝ,h_{p,W,\varphi }(q):=\{\begin{array}{cc}\eta (\frac{1}{ϵ}(\varphi (p)-\varphi (q)))& q\in O\\ 0& q\notin O\end{array}$,

then we have $h_{p,W,\varphi }\in {𝒞}^n(M)$.

Proof:

Let $(U,\theta )\in \{(O_{\upsilon },{\varphi }_{\upsilon })|\upsilon \in \mathrm{{\rm Y}}\}$. Then we have for $x\in \theta (U)$:

$(h_{p,W,\varphi }{|}_U\circ {\theta }^{-1})(x)=\{\begin{array}{cc}0& (\varphi \circ {\theta }^{-1})(x)\notin B_{ϵ}(\varphi (p))\\ e^{1-\frac{1}{1-\Vert \frac{1}{ϵ}(\varphi (p)-(\varphi \circ {\theta }^{-1})(x)){\Vert }^2}}& (\varphi \circ {\theta }^{-1})(x)\in B_{ϵ}(\varphi (p))\end{array}$

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 $\{V_{\kappa }|\kappa \in \mathrm{K}\}$ be an open cover of $M$.

We choose for each point $p\in M$ an atlas $(O,\varphi )$ 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 $h_{p,V_{{\kappa }_p}\cap O,\varphi }\in {𝒞}^n(M)$ such that $p\in \{q\in M|h_{p,V_{{\kappa }_p}\cap O,\varphi }(q)>0\}=:W_p$. Since $h_{p,V_{{\kappa }_p}\cap O,\varphi }$ is continuous, all the $W_p$ are open; this is because they are preimages of the open set $(0,\mathrm{\infty })$. 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

$\phi :M\to ℝ,\phi (q):=\sum _{W_p\in S}{h}_{p,V_{{\kappa }_p}\cap O,\varphi }(q)$

This function is of class ${𝒞}^n(M)$ 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 ${𝒞}^n(M)$ functions (that finite sums of ${𝒞}^n(M)$ functions are again ${𝒞}^n(M)$ 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 ${\phi }_{p,V_{{\kappa }_p}\cap O,\varphi }:=\frac{h_{p,V_{{\kappa }_p}\cap O,\varphi }}{\phi }$ are contained in ${𝒞}^n(M)$. 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.$◻$

Sources

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