Vector Bundles

Lee Smooth Manifolds Owing to their extreme abstractness, a unified approach to representing and interpreting smooth vector bundles over manifolds is greatly helpful; this is precisely the goal of these notes. Understanding the construction theorems from the ground up is a good starting point.

Theorem 1 (Lee 1.35): Let $M$ be a set, and suppose we are given a collection $\{U_{\alpha }\}$ of subsets of $M$ together with maps

$${\phi }_{\alpha }:U_{\alpha }\to {\mathbb{R}}^n$$

such that the following properties are satisfied: (i) For each $\alpha$, ${\phi }_{\alpha }$ is a bijection between $U_{\alpha }$ and an open subset ${\phi }_{\alpha }(U_{\alpha })\subseteq {\mathbb{R}}^n$. (ii) For each $\alpha$ and $\beta$, the sets ${\phi }_{\alpha }(U_{\alpha }\cap U_{\beta })$ and ${\phi }_{\beta }(U_{\alpha }\cap U_{\beta })$ are open in ${\mathbb{R}}^n$. (iii) Whenever $U_{\alpha }\cap U_{\beta }\ne \varnothing$, the map

$${\phi }_{\beta }\circ {\phi }_{\alpha }^{-1}:{\phi }_{\alpha }(U_{\alpha }\cap U_{\beta })\to {\phi }_{\beta }(U_{\alpha }\cap U_{\beta })$$

is smooth. (iv) Countably many of the sets $U_{\alpha }$ cover $M$. (v) Whenever $p,q$ are distinct points in $M$, either there exists some $U_{\alpha }$ containing both $p$ and $q$, or there exist disjoint sets $U_{\alpha },U_{\beta }$ with $p\in U_{\alpha }$ and $q\in U_{\beta }$. Then $M$ has a unique smooth manifold structure such that each $(U_{\alpha },{\phi }_{\alpha })$ is a smooth chart.

Proof) Define a topological basis $\mathcal{B}$ on $M$ as follows: $U\in \mathcal{B}$ whenever there is some $U\subset U_{\alpha }$ such that ${\phi }_{\alpha }(U)$ is open in ${\mathbb{R}}^n$. We define the topology on $M$ as the one generated by $\mathcal{B}$; one can quickly verify that, with this topology, every ${\phi }_{\alpha }$ is a homeomorphism and $U_{\alpha }$ is open in $M$ (using (i), (ii), and (iii)). Thus, $M$ is locally Euclidean, and, by (iv), second-countable. By (v), $M$ is Hausdorff; therefore, $M$ is a topological manifold.

On top of the topology of $M$, the collection $\{(U_{\alpha },{\phi }_{\alpha })\}$ is also a smooth atlas by (iii), which determines a smooth structure on $M$ such that each $(U_{\alpha },{\phi }_{\alpha })$ is a smooth chart. Uniqueness is trivial.

Remark: The same theorem applies to manifolds with boundary if we change ${\mathbb{R}}^n$ to ${\mathbb{H}}^n$ in the first equation. The proof is identical.

Theorem 2 (Lee 10.6): Let $M$ be a smooth manifold with or without boundary, and suppose that for each $p\in M$ we are given a real vector space $E_p$ of some fixed dimension $k$. Let

$$E=\coprod _{p\in M}{E}_p,$$

and let

$$\pi :E\to M$$

be the map that takes each element of $E_p$ to the point $p$. Suppose furthermore that we are given the following data: (i) an open cover $\{U_{\alpha }{\}}_{\alpha \in A}$ of $M$. (ii) for each $\alpha \in A$, a bijective map ${\Phi }_{\alpha }:{\pi }^{-1}(U_{\alpha })\to U_{\alpha }\times {\mathbb{R}}^k$ whose restriction to each $E_p$ is a vector space isomorphism from $E_p$ to $p\times {\mathbb{R}}^k\cong {\mathbb{R}}^k$. (iii) for each $\alpha ,\beta \in A$ with $U_{\alpha }\cap U_{\beta }\ne \varnothing$, a smooth map ${\tau }_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to \mathrm{GL}(k,\mathbb{R})$ such that the map ${\Phi }_{\alpha }\circ {\Phi }_{\beta }^{-1}$ from $(U_{\alpha }\cap U_{\beta })\times {\mathbb{R}}^k$ to itself has the form ${\Phi }_{\alpha }\circ {\Phi }_{\beta }^{-1}(p,v)=(p,{\tau }_{\alpha \beta }(p)v)$. Then $E$ has a unique topology and smooth structure making it into a smooth manifold with or without boundary and a smooth rank-$k$ vector bundle over $M$, with $\pi$ as projection and $(U_{\alpha },{\Phi }_{\alpha })$ as smooth local trivializations.

Proof) For every $p\in U_{\alpha }$, take all smooth charts $(V_p,{\varphi }_p)$ such that $p\in V_p\subset U_{\alpha }$; we consider the set ${\pi }^{-1}(V_p)\subset E$ together with ${\phi }_p=({\varphi }_p\times I{d}_{{\mathbb{R}}^k})\circ {\Phi }_{\alpha }$ which is a bijection from ${\pi }^{-1}(V_p)$ to the open set ${\varphi }_p(V_p)\times {\mathbb{R}}^k$ in ${\mathbb{R}}^{m+k}$ where $m=\text{dim}M$, and claim Theorem 1 is applicable. Clearly, (i) is satisfied. (iv) is satisfied as countably many such sets $V_p$ cover $M$, implying countably many sets ${\pi }^{-1}(V_p)$ cover $E$, and (ii) is satisfied by the smoothness of $M$. (iii) is true as

$${\phi }_p\circ {\phi }_q^{-1}=({\varphi }_p\times I{d}_{{\mathbb{R}}^k})\circ {\Phi }_{\alpha }\circ {\Phi }_{\beta }^{-1}\circ ({\varphi }_q\times I{d}_{{\mathbb{R}}^k}{)}^{-1}$$

which is a diffeomorphism as $({\varphi }_p\times I{d}_{{\mathbb{R}}^k})$, ${\Phi }_{\alpha }\circ {\Phi }_{\beta }^{-1}$, and $({\varphi }_q\times I{d}_{{\mathbb{R}}^k}{)}^{-1}$ are. (v) is also trivial after splitting into cases where $e_1,e_2\in E$ are in different or the same fiber of $\pi$. We thus obtain a smooth manifold structure on $E$ such that all applicable $({\pi }^{-1}(V_p),{\phi }_p)$ are smooth charts. Each $(U_{\alpha },{\Phi }_{\alpha })$ is then a smooth trivialization as in each $V_p$ it has the coordinate representation of the identity map. It follows that $E$ is a vector bundle over $M$ with $\pi$ as the projection and $(U_{\alpha },{\Phi }_{\alpha })$ as smooth local trivializations; uniqueness is trivial.

Theorem 3 (Lee E10.6): Let $M$ be a smooth manifold with or without boundary, and let $\{U_{\alpha }{\}}_{\alpha \in A}$ be an open cover of $M$. Suppose for each $\alpha ,\beta \in A$ we are given a smooth map ${\tau }_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to \mathrm{GL}(k,\mathbb{R})$ such that the identity

$${\tau }_{\alpha \beta }(p){\tau }_{\beta \gamma }(p)={\tau }_{\alpha \gamma }(p),\forall p\in U_{\alpha }\cap U_{\beta }\cap U_{\gamma }$$

is satisfied for all $\alpha ,\beta ,\gamma \in A$. Show that there is a smooth rank-$k$ vector bundle $E\to M$ with smooth local trivializations

$${\Phi }_{\alpha }:{\pi }^{-1}(U_{\alpha })\to U_{\alpha }\times {\mathbb{R}}^k$$

whose transition functions are the given maps ${\tau }_{\alpha \beta }$.

Proof) Consider the set $\bigcup (U_{\alpha }\times {\mathbb{R}}^k)$ with an equivalence relation $\sim$ given by $(p,v)\in U_{\alpha }\times {\mathbb{R}}^k\sim (q,w)\in U_{\beta }\times {\mathbb{R}}^k$ iff $p=q$ and $v={\tau }_{\alpha \beta }(p)w$; it is a simple exercise to show $\sim$ is an equivalence relation. Now let $E$ be the quotient set $\bigcup (U_{\alpha }\times {\mathbb{R}}^k)/\sim$ with the canonical projection $\pi$. Each fiber of $\pi$ becomes a well-defined vector space, allowing us to apply Theorem 2 with $(U_{\alpha },{\Phi }_{\alpha })$ where, for each $[p]\in {\pi }^{-1}(U_{\alpha })$, ${\Phi }_{\alpha }([p])=p\in U_{\alpha }\times {\mathbb{R}}^k$ where $p$ is the representative of the equivalence class belonging to $U_{\alpha }\times {\mathbb{R}}^k$. This clearly satisfies Theorem 2, giving us a unique smooth structure on $E$ such that it is a vector bundle with $(U_{\alpha },{\Phi }_{\alpha })$ as smooth trivializations.

Theorem 4 (Lee E10.12): Let $\pi :E\to M$ and $\tilde{\pi }:\tilde{E}\to M$ be two smooth rank-$k$ vector bundles over a smooth manifold $M$ with or without boundary. Suppose $\{U_{\alpha }{\}}_{\alpha \in A}$ is an open cover of $M$ such that both $E$ and $\tilde{E}$ admit smooth local trivializations over each $U_{\alpha }$. Let ${\tau }_{\alpha \beta }$ and ${\tilde{\tau }}_{\alpha \beta }$ denote the transition functions determined by the given local trivializations of $E$ and $\tilde{E}$, respectively. Show that $E$ and $\tilde{E}$ are smoothly isomorphic over $M$ if and only if for each $\alpha \in A$ there exists a smooth map ${\sigma }_{\alpha }:U_{\alpha }\to \mathrm{GL}(k,\mathbb{R})$ such that

$${\tilde{\tau }}_{\alpha \beta }(p)={\sigma }_{\alpha }(p){\tau }_{\alpha \beta }(p){\sigma }_{\beta }(p{)}^{-1},\forall p\in U_{\alpha }\cap U_{\beta }$$

Proof) Suppose $E$ and $\tilde{E}$ are smoothly isomorphic with isomorphism $\phi :E\to \tilde{E}$. For each $p\in U_{\alpha }$ if we let ${\Phi }_{\alpha }$ be the trivialization over $U_{\alpha }$ in $E$ and ${\tilde{\Phi }}_{\alpha }$ be the trivialization in $\tilde{E}$, ${\tilde{\Phi }}_{\alpha }\circ \phi \circ {\Phi }_{\alpha }^{-1}$ determines a vector space isomorphism from $E_p$ to ${\tilde{E}}_p$ that we represent with matrix ${\sigma }_{\alpha }(p)\in GL(k,\mathbb{R})$; it is not difficult to see ${\sigma }_{\alpha }$ is a smooth map from $U_{\alpha }$ to $GL(k,\mathbb{R})$. Then, ${\tilde{\tau }}_{\alpha \beta }(p)={\sigma }_{\alpha }(p){\tau }_{\alpha \beta }(p){\sigma }_{\beta }(p{)}^{-1}$ and we are done.

For the opposite direction, take $\phi :E\to \tilde{E}$ such that for any $p\in U_{\alpha }$, under the coordinate representation induced by the smooth trivializations on $U_{\alpha }$,

$$\phi (p,v)=(p,{\sigma }_{\alpha }(p)v)$$

for any $v\in {\mathbb{R}}^k$. This is well-defined as, for any $p\in U_{\alpha }\cap U_{\beta }$,

$${\sigma }_{\alpha }(p){\tau }_{\alpha \beta }(p)v={\tilde{\tau }}_{\alpha \beta }(p){\sigma }_{\beta }(p)v$$

One can clearly see $\phi$ is a bijective smooth bundle homomorphism over $M$, and thus is a smooth bundle isomorphism.

Theorem 5: Let $M$ be a smooth manifold with or without boundary and $\pi :E\to M$ be a rank $k$ smooth vector bundle. Let $\{U_{\alpha }{\}}_{\alpha \in A}$ be an open cover of $M$ such that there are smooth trivializations ${\varphi }_{\alpha }:{\pi }^{-1}(U_{\alpha })\to U_{\alpha }\times {\mathbb{R}}^k$ with transition functions ${\tau }_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to GL(k,\mathbb{R})$. Define $\tilde{E}$ as the adjunction set ${\coprod }_{\sim }{U}_{\alpha }\times {\mathbb{R}}^k$ where $(p,v)\in U_{\alpha }\times {\mathbb{R}}^k\sim (q,w)\in U_{\beta }\times {\mathbb{R}}^k$ iff $p=q$ and $w={\tilde{\tau }}_{\beta \alpha }v$. There is a smooth structure on $\tilde{E}$ such that, with the canonical projection $\tilde{\pi }$, $\tilde{E}$ is a rank $k$ smooth vector bundle over $M$ smoothly isomorphic to $E$ over $M$.

Proof) We give $\tilde{E}$ the canonical manifold structure from Theorem 3; the cyclic identity is clearly satisfied as $E$ is a valid vector bundle. We then have that $\tilde{E}$ is smoothly isomorphic to $E$ by Theorem 4 with identity maps ${\sigma }_{\alpha }:p\mapsto I_k$.

Remark: This theorem shows that, up to smooth bundle isomorphisms, the structure of a smooth vector bundle is entirely determined by a set of smooth trivializations that cover the base manifold. It also shows all vector bundles can be intuitively understood as the union of multiple product spaces.