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

Maximal atlases

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 $A$.$\Box$

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.$\Box$

Theorem 3.4:

Let $M$ be a $d$-dimensional manifold with atlas $\{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$, and let $A$ be it's maximal atlas. There does not exist an atlas $B$ such that $A\subsetneq B$ (this notation shall mean that $A$ is contained in $B$, but $B$ is 'strictly larger than $A$' (by this obscure saying we shall mean that there exists at least one element in $B$ which is not contained in $A$)).

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!$\Box$

Second-countable spaces

Definition 3.5:

Let $M$ be a topological space and let $U_{\kappa },\kappa \in \mathrm {K}$ be a set of open sets. We call $\{U_{\kappa }|\kappa \in \mathrm {K} \}$ a basis of the topology of $M$ iff every open set $O\subseteq M$ can be written as the union of elements of $\{U_{\kappa }|\kappa \in \mathrm {K} \}$, i. e.

$O=\bigcup _{\upsilon \in \Upsilon }U_{\upsilon }$

where $\Upsilon \subseteq \mathrm {K}$.

Definition 3.6:

Let $M$ be a topological space. We call $M$ second-countable iff $M$'s topology has a countable basis.

Locally finite refinements and partitions of unity

Definition 3.7:

Let ${\mathcal {X}}$ be a topological space. An open cover of ${\mathcal {X}}$ is a set $\{V_{\kappa }|\kappa \in \mathrm {K} \}$ of open subsets of ${\mathcal {X}}$ such that

$\bigcup _{\kappa \in \mathrm {K} }V_{\kappa }={\mathcal {X}}$

Example 3.8:

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

Definition 3.9:

Let $M$ be a manifold of class ${\mathcal {C}}^{k}$. We say that $M$ admits partitions of unity iff for every open cover $\{V_{\kappa }|\kappa \in \mathrm {K} \}$ there exist functions $\{\varphi _{\iota }|\iota \in \mathrm {I} \}\subset {\mathcal {C}}^{n}(M)$ such that:

1. For all $\iota \in \mathrm {I}$, there exists a $\kappa \in \mathrm {K}$ such that ${\text{supp }}\varphi _{\iota }\subset V_{\kappa }$,
2. for all $\iota \in \mathrm {I}$ and $p\in M$, $\varphi _{\iota }(p)\geq 0$ AND
3. for all $p\in M$, $\sum _{\iota \in \mathrm {I} }\varphi _{\iota }(p)=1$.

Definition 3.10:

Let ${\mathcal {X}}$ be a topological space and let $\{V_{\kappa }|\kappa \in \mathrm {K} \}$ be an open cover of ${\mathcal {X}}$. A locally finite refinement of $\{V_{\kappa }|\kappa \in \mathrm {K} \}$ is defined to be another open cover of ${\mathcal {X}}$, say $\{U_{\upsilon }|\upsilon \in \Upsilon \}$, such that:

• for each $\upsilon \in \Upsilon$, there exists a $\kappa \in \mathrm {K}$ such that $U_{\upsilon }\subseteq V_{\kappa }$, AND
• for each $x\in {\mathcal {X}}$, the set $\{U_{\upsilon }|\upsilon \in \Upsilon \wedge x\in V_{\upsilon }\}$ is finite.

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 $\{V_{\kappa },\kappa \in \mathrm {K} \}$ is an cover of $\phi ^{-1}({\overline {W_{p}}})$ by open subsets of $M$, then the set $\{V_{\kappa }\cap O,\kappa \in \mathrm {K} \}$ 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 $\{V_{\kappa },\kappa \in \mathrm {K} \}$. 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.$\Box$

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 $l\geq 2$ (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 $l\geq 3$ (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.$\Box$

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.$\Box$

Theorem 3.14:

Let $M$ be a manifold of class ${\mathcal {C}}^{n}$ with a countable basis, i. e. a second-countable manifold. Then $M$ admits partitions of unity.

Proof:

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

We choose for each point $p\in M$ an atlas $(O,\phi )$ 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.$\Box$

Sources

• Lang, Serge (2002). Introduction to Differentiable Manifolds. New York: Springer. ISBN 0-387-95477-5.