Vector Bundles

Vector Bundle

A vector bundle of rank $k$ over a base topological space $X$ (usually a smooth manifold) consists of the following information:

• topological spaces $X$ and $E$;
• a continuous surjection $\pi :E\to X$, called the bundle projection;
• for every $x\in X$, the fiber ${\pi }^{-1}(\{x\})$ yields a finite-dimensional real vector space,

such that $\pi$ is locally trivialisable. That is, for every point $p\in X$, there is an open neighbourhood $U\subseteq X$ of $p$, a natural number $k$, and a homeomorphism \newcommand{\R}{\mathbb{R}}\Phi \colon \pi^{-1}(U) \to U \times \R^{k} that for all $x\in U$, $\Phi {|}_{{\pi }^{-1}(\{x\})}$ is a linear isomorphism onto $\{x\}\times {\mathbb{R}}^k$. In other words, the following diagram commutes:

Remark

Intuitively, the local trivialisation means that locally, the vector bundle doesn’t twist or bend in a complicated way. So, local trivialization is the property that allows us to study the vector bundle piece by piece, using the familiar structure of product spaces in each piece, even if the entire bundle has a more complex, “twisted” global structure.

Transition Map

Section

A section $s:X\to E$ of a vector bundle $E$ over $X$ through $\pi$ is essentially an assignment to choose one vector from each fiber of the bundle in a smooth manner, such that $\pi \circ s=\text{id}$.

e.g. A vector bundle $E$ over a manifold $M$ with section $s$ and fibers $E_{m_i}$:

Proposition

Let $\pi :E\to X$ be a vector bundle. The space of sections, denoted $\Gamma (E)$, is a vector space, with a module structure over the ring $C^{\infty }(M)$.

Frame of a Vector Bundle

Let $E$ be a vector bundle on a smooth manifold $M$. A local frame over an open subset $U\subset M$, is a set of $k$ smooth sections $\{s_1,s_2,\dots ,s_k\}$ of $E$ such that for every $x\in U$, the vectors $\{s_1(x),s_2(x),\dots ,s_k(x)\}$ form a basis of the fiber $E_x$.

Remark

A frame provides a way to trivialise the bundle locally, giving a smooth choice of basis for each vector space fiber over an open set of the base manifold.

On an $n$-manifold, given two coordinate charts ${\phi }_{\alpha }=(x_{\alpha }^1,x_{\alpha }^2,\dots ,x_{\alpha }^n):U_{\alpha }\to {\mathbb{R}}^n,{\phi }_{\beta }=(x_{\beta }^1,x_{\beta }^2,\dots ,x_{\beta }^n):U_{\beta }\to {\mathbb{R}}^n,$the coordinate change ${\phi }_{\alpha }\circ {\phi }_{\beta }^{-1}:{\phi }_{\beta }(U_{\alpha }\cap U_{\beta })\to {\phi }_{\alpha }(U_{\alpha }\cap U_{\beta })$induces a corresponding change of basis: $\left(\begin{array}{c}\text{pddf}{x}_{\alpha }^1\\ \text{pddf}{x}_{\alpha }^2\\ ⋮\\ \text{pddf}{x}_{\alpha }^n\end{array}\right)=\left(\begin{array}{cccc}\frac{\partial x_{\beta }^1}{\partial x_{\alpha }^1}& \frac{\partial x_{\beta }^2}{\partial x_{\alpha }^1}& \cdots & \frac{\partial x_{\beta }^n}{\partial x_{\alpha }^1}\\ \frac{\partial x_{\beta }^1}{\partial x_{\alpha }^2}& \frac{\partial x_{\beta }^2}{\partial x_{\alpha }^2}& \cdots & \frac{\partial x_{\beta }^n}{\partial x_{\alpha }^2}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial x_{\beta }^1}{\partial x_{\alpha }^n}& \frac{\partial x_{\beta }^2}{\partial x_{\alpha }^n}& \cdots & \frac{\partial x_{\beta }^n}{\partial x_{\alpha }^n}\end{array}\right)\left(\begin{array}{c}\text{pddf}{x}_{\beta }^1\\ \text{pddf}{x}_{\beta }^2\\ ⋮\\ \text{pddf}{x}_{\beta }^n\end{array}\right).$And we can identify the change of basis matrix ${\left(\text{pddf}{x}_{\beta }^i{x}_{\alpha }^j\right)}_{n\times n}$ as a smooth map $U_{\alpha }\cap U_{\beta }\to \mathrm{GL}(n,\mathbb{R})$.

Subbundle

$F\to X$ is called a subbundle of $E\to X$ if for every $x\in X$, the fiber $F_x$ is a linear subspace of the fiber $E_x$.

Quotient Bundle

Given a subbundle $F\to X$ of a vector bundle $E\to X$, the quotient bundle $E/F$ is the vector bundle over $X$ whose fibers are the quotient spaces $E_x/F_x$, for all $x\in X$.

Dual Bundle

The dual bundle of a vector bundle $E$ over $X$ is the vector bundle $E^{\ast }$ over $X$ whose fibers are the dual spaces to the fibers of $E$.

Pullback Bundle & Pullback Section

Suppose $\pi :E\to N$ is a vector bundle on a manifold $N$, and $F:M\to N$ is a local diffeomorphism map. Then the pullback bundle $F^{\ast }E$ is a vector bundle over $M$ whose fibers are the pullbacks of the fibers of $E$. That is, for each $p\in M$, the fiber $(F^{\ast }E{)}_p$ is defined as $(F^{\ast }E{)}_p:=E_{F(p)}$. A section $s$ of $E$ induces a section $F^{\ast }s$ of $F^{\ast }E$ by $F^{\ast }s(p):=s(F(p)).$

Lemma

A continuous map $h:E_1\to E_2$ between vector bundles over the same base space $B$ is an isomorphism if it takes each fiber $p_1^{-1}(b)$ to the corresponding fiber $p_2^{-1}(b)$ by a linear isomorphism.

Proposition

The only spheres with trivial tangent bundles are $S^1,S^3,S^7$.

Direct Sum of Vector Bundles

Given two vector bundles $p_1:E_1\to B$ and $p_2:E_2\to B$ over the same base space $B$, the direct sum of $E_1$ and $E_2$ is defined as the space $E_1\oplus E_2=\{(v_1,v_2)\in E_1\times E_2\mid p_1(v_1)=p_2(v_2)\}$ There is then a projection $E_1\oplus E_2\to B$ sending $(v_1,v_2)$ to the point $p_1(v_1)=p_2(v_2)$.

Remark

The direct sum of two trivial bundles is again a trivial bundle, clearly, but the direct sum of nontrivial bundles can also be trivial. For example consider the direct sum of tangent and normal bundle to $S^n$ in ${\mathbb{R}}^{n+1}$.

Inner Product on a Vector Bundle

An inner product on a vector bundle $p:E\to B$ is a map $⟨\cdot ,\cdot ⟩:E\oplus E\to \mathbb{R}$ which restricts in each fiber to an inner product, a positive definite symmetric bilinear form.

$p:E\to B$ if $B$ is compact Hausdorff or more generally paracompact.

An inner product exists for a vector bundle

Proposition

If $E\to B$ is a vector bundle over a paracompact base $B$ and $E_0\subseteq E$ is a vector subbundle, then there is a vector subbundle $E_0^{\perp }\subseteq E$ such that $E_0\oplus E_0^{\perp }\cong E$.

Tensor Product Bundle

Let $p_1:E_1\to B$ and $p_2:E_2\to B$ be vector bundles. The tensor product bundle $E_1\otimes E_2\to B$ is defined by: $E_1\otimes E_2={\coprod }_{x\in B}(E_1{)}_x\otimes (E_2{)}_x$ $\text{Fiber: }(E_1\otimes E_2{)}_x=(E_1{)}_x\otimes (E_2{)}_x$ A cover $\{U_{\alpha }\}$ of $B$ and homeomorphisms $h_{\alpha }:(E_1\otimes E_2){|}_{U_{\alpha }}\stackrel{\cong }{\to }{U}_{\alpha }\times {\mathbb{R}}^{n_1{n}_2}$ induced by local trivializations of $E_1\text{ and }{E}_2$.