# General Geometry/Manifolds

Definition (manifold):

Let ${\displaystyle (C,G)}$ be a site, and let ${\displaystyle D}$ be a subcategory of ${\displaystyle C}$. A manifold of type ${\displaystyle D}$ consists of the aforementioned site ${\displaystyle (C,G)}$ together with a class ${\displaystyle (\varphi _{\alpha }:U_{\alpha }\to X_{\alpha })_{\alpha }}$ of isomorphisms of ${\displaystyle C}$, where the ${\displaystyle U_{\alpha }}$'s are open in ${\displaystyle G}$ and the ${\displaystyle X_{\alpha }}$ are objects of ${\displaystyle D}$, such that

1. for all ${\displaystyle U\in \operatorname {Obj} (C)}$ there is a covering ${\displaystyle \iota _{i}:U_{i}\to U}$ of ${\displaystyle U}$ such that for each ${\displaystyle i}$, there exists an ${\displaystyle \alpha }$ with ${\displaystyle U_{i}=U_{\alpha }}$ and
2. whenever ${\displaystyle U,V,W\in \operatorname {Obj} C}$ st. ${\displaystyle \iota _{1}:U\to W}$ and ${\displaystyle \iota _{2}:V\to W}$ consitute a covering and ${\displaystyle \varphi :U\to X}$ and ${\displaystyle \psi :V\to Y}$ belong to the class ${\displaystyle (\varphi _{\alpha }:U_{\alpha }\to X_{\alpha })_{\alpha }}$, the maps ${\displaystyle \varphi \upharpoonright (U\times _{W}V)}$ and ${\displaystyle \psi \upharpoonright (U\times _{W}V)}$ guaranteed by the universal property of the pullback are isomorphisms onto their respective images and ${\displaystyle (\psi \upharpoonright (U\times _{W}V))\circ (\varphi \upharpoonright (U\times _{W}V))^{-1}}$ is a morphism of ${\displaystyle D}$.

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Proposition (fundamental induction lemma):

Let ${\displaystyle P(M,A)}$  be a logical statement, whose arguments are a topological manifold ${\displaystyle M}$  and a closed subset ${\displaystyle A\subseteq M}$ . Suppose that the following are true:

1. Whenever ${\displaystyle B\subseteq V_{\alpha }}$  is compact and convex, where ${\displaystyle \phi _{\alpha }:U_{\alpha }\to V_{\alpha }}$  is a coördinate chart on ${\displaystyle M}$ , then ${\displaystyle P(M,\phi _{\alpha }^{-1}(B))}$  is true
2. Whenever ${\displaystyle P(M,A),P(M,B)}$  and ${\displaystyle P(M,A\cap B)}$  are true, then ${\displaystyle P(M,A\cup B)}$  is true
3. Whenever ${\displaystyle A_{1}\supseteq A_{2}\supseteq A_{3}\supseteq \cdots }$  is a descending chain of compact subsets of ${\displaystyle M}$ , ${\displaystyle P(M,A_{j})}$  is true for all ${\displaystyle j\in \mathbb {N} }$ , then ${\displaystyle P\left(M,\bigcap _{j\in \mathbb {N} }A_{j}\right)}$  is true
4. Whenever ${\displaystyle P(M,A\cap {\overline {W}})}$  is true for all relatively compact, open ${\displaystyle W\subseteq M}$ , then ${\displaystyle P(M,A)}$  is true

Then for all closed ${\displaystyle A\subseteq M}$ , ${\displaystyle P(M,A)}$  is true.

Proof: First we prove by induction on ${\displaystyle k}$  that for all sets of the type ${\displaystyle A_{j}=\phi _{\alpha }^{-1}(B_{j})}$  for certain compact convex ${\displaystyle B_{j}}$  (${\displaystyle j\in \{1,\ldots ,k\}}$ ), the statement ${\displaystyle P(M,A_{1}\cup \cdots \cup A_{k})}$  is true. We proceed by induction on ${\displaystyle k}$ . For ${\displaystyle k=1}$ , the statement is implied by the first assumption. Suppose now that ${\displaystyle P(M,A_{1}\cup \cdots \cup A_{k-1})}$  is true. Note that also ${\displaystyle P(M,A_{k})}$  is true by the first assumption. Also

${\displaystyle (A_{1}\cup \cdots \cup A_{k-1})\cap A_{k}=(A_{1}\cap A_{k})\cup \cdots \cup (A_{k-1}\cap A_{k})}$ ,

and ${\displaystyle A_{j}\cap A_{k}=\phi _{\alpha }^{-1}(B_{j}\cap B_{k})}$ , where ${\displaystyle B_{j}\cap B_{k}}$  is compact and convex as the intersection of two compact and convex sets. Thus, since the sets ${\displaystyle A_{1}\cap A_{k},\ldots ,A_{1}\cap A_{k-1}}$  are only ${\displaystyle k-1}$  many, by induction on ${\displaystyle k}$ , we may also conclude that ${\displaystyle P(M,(A_{1}\cup \cdots \cup A_{k-1})\cap A_{k})}$  holds. By 2., we conclude ${\displaystyle P(M,A_{1}\cup \cdots \cup A_{k})}$ .

Now, we prove that ${\displaystyle P(M,A)}$  is true whenever ${\displaystyle A}$  is a compact subset of a set of the type ${\displaystyle \phi _{\alpha }^{-1}(B)}$ , where ${\displaystyle B}$  is compact and convex. Indeed, for each ${\displaystyle m\in \mathbb {N} }$ , cover ${\displaystyle \phi _{\alpha }(A)}$  by all the cubes of sidelength ${\displaystyle 2/2^{m}}$  centered at the points of ${\displaystyle (1/2^{m})\mathbb {Z} ^{n}}$  that intersect it. Then set

${\displaystyle A_{m}:=\bigcup _{j=1}^{k_{m}}B_{j,m}}$ , so that ${\displaystyle A=\bigcap _{m\in \mathbb {N} }A_{m}}$ .

By the second assumption, ${\displaystyle P(M,A_{m})}$  holds for each ${\displaystyle m\in \mathbb {N} }$ , and hence by the third assumption ${\displaystyle P(M,A)}$  holds.

Now we claim by induction on ${\displaystyle k}$  that whenever ${\displaystyle A_{1},\ldots ,A_{k}}$  are compact subsets of sets of the type ${\displaystyle \phi _{\alpha }^{-1}(B)}$  (${\displaystyle B}$  compact and convex), then ${\displaystyle P(M,A_{1}\cup \cdots \cup A_{k})}$  holds. For ${\displaystyle k=1}$ , this follows from what we just proved. For the induction step, suppose that ${\displaystyle P(M,A_{1}\cup \cdots \cup A_{k-1})}$  holds. Note that also ${\displaystyle P(M,A_{k})}$  holds by what we just proved. Then we have

${\displaystyle (A_{1}\cup \cdots \cup A_{k-1})\cap A_{k}=(A_{1}\cup A_{k})\cup \cdots \cup (A_{k-1}\cap A_{k})}$ ,

and since ${\displaystyle A_{j}\cap A_{k}}$  is a compact subset of ${\displaystyle A_{j}}$ , ${\displaystyle P(M,A_{j}\cap A_{k})}$  holds by what we just proved, and therefore, by induction, ${\displaystyle P(M,(A_{1}\cup A_{k})\cup \cdots \cup (A_{k-1}\cap A_{k})}$  holds. Hence, by 2., we get that ${\displaystyle P(M,A_{1}\cup \cdots \cup A_{k})}$  is true.

Now we are ready to prove that ${\displaystyle P(M,A)}$  is true whenever ${\displaystyle A}$  is compact. Indeed, let ${\displaystyle A\subseteq M}$  be compact. Then cover ${\displaystyle A}$  by sets ${\displaystyle \phi _{\alpha }^{-1}(B_{\alpha })}$ , where ${\displaystyle \phi _{\alpha }}$  are some charts and ${\displaystyle B_{\alpha }}$  is compact and convex. By compactness of ${\displaystyle A}$ , we may extract a finite subcover ${\displaystyle \phi _{\alpha _{1}}^{-1}(B_{\alpha _{1}})=:A_{1},\ldots ,A_{n}:=\phi _{\alpha _{n}}^{-1}(B_{\alpha _{n}})}$ . By intersecting with ${\displaystyle A}$  and retaining compactness (as the intersection of two compact sets is compact), we may assume that ${\displaystyle A_{1},\ldots ,A_{n}}$  are contained within ${\displaystyle A}$  and in particular

${\displaystyle A=\bigcup _{j=1}^{n}A_{j}}$ .

Thus, by the previous step, ${\displaystyle P(M,A)}$  holds.

Finally, let ${\displaystyle A}$  be an arbitrary closed subset of ${\displaystyle M}$ , and let ${\displaystyle W\subseteq M}$  be a relatively compact open subset. Then ${\displaystyle P(M,A\cap {\overline {W}})}$  is true since ${\displaystyle A\cap {\overline {W}}}$  is compact as a closed subset of a compact set. Hence, by the fourth assumption, ${\displaystyle P(M,A)}$  is true. ${\displaystyle \Box }$

Definition (vector bundle):

Let ${\displaystyle R}$  be a topological ring, and let ${\displaystyle M}$  be a manifold. An ${\displaystyle n}$ -dimensional ${\displaystyle R}$ -vector bundle over ${\displaystyle M}$  is a manifold ${\displaystyle E}$  together with a morphism of manifolds ${\displaystyle \pi :E\to M}$  such that:

1. for each ${\displaystyle x\in M}$ , the set ${\displaystyle E_{x}:=\pi ^{-1}(x)}$  (which is called the fibre of ${\displaystyle x}$ ), is a finite-dimensional topological vector space over ${\displaystyle R}$
2. for each ${\displaystyle x\in M}$ , there exists a neighbourhood ${\displaystyle U}$  of ${\displaystyle x}$ , an ${\displaystyle n\in \mathbb {N} }$  and a map ${\displaystyle \Psi :\pi ^{-1}(U)\to U\times R^{n}}$  which is a fibre-wise TVS isomorphism such that the diagram

commutes.