9.3 Vector bundles

9.3.1 Definitions and examples

A vector bundle $E\stackrel{\pi }{\to }M$ is a fibre bundle whose fibre is a vector space. Let $F$ be ${\mathbb{R}}^k$ and $M$ be an $m$ -dimensional manifold. It is common to call $k$ the fibre dimension and denote it by $\dim E$ , although the total space $E$ is $m+k$ dimensional. The transition functions belong to $\mathrm{GL}(k,\mathbb{R})$ , since it maps a vector space onto another vector space of the same dimension isomorphically. If $F$ is a complex vector space ${\mathbb{C}}^k$ , the structure group is $\mathrm{GL}(k,\mathbb{C})$ .

Example 9.3

A tangent bundle $TM$ over an $m$ -dimensional manifold $M$ is a vector bundle whose typical fibre is ${\mathbb{R}}^m$ , see section 9.1. Let $u$ be a point in $TM$ such that $\pi (u)=p\in U_i\cap U_j$ , where $\{U_i\}$ covers $M$ . Let $x^{\mu }={\phi }_i(p)$

$(y^{\mu }={\phi }_j(p))$ be the coordinate system of $U_i$ $(U_j)$ . The vector $V$ corresponding to $u$ is expressed as $V=V^{\mu }\partial /\partial x^{\mu }{|}_p={\tilde{V}}^{\mu }\partial /\partial y^{\mu }{|}_p$ . The local trivializations are

$${\varphi }_i^{-1}(u)=\left(p,\left\{V^{\mu }\right\}\right){\varphi }_j^{-1}(u)=\left(p,\left\{{\tilde{V}}^{\mu }\right\}\right).\text{\hspace{1em}(9.20)}$$

The fibre coordinates $\{V^{\mu }\}$ and $\{{\tilde{V}}^{\mu }\}$ are related as

$$V^{\mu }=G_v^{\mu }(p){\tilde{V}}^v\text{\hspace{1em}(9.21)}$$

where $\{G_{\nu }^{\mu }(p)\}=\{(\partial x^{\mu }/\partial y^{\nu }{)}_p\}\in \mathrm{GL}(m,\mathbb{R})$ . Hence, a tangent bundle is $(TM,\pi ,M,{\mathbb{R}}^m,\mathrm{GL}(m,\mathbb{R}))$ . Sections of $TM$ are the vector fields on $M$ ; $\mathcal{X}(M)=\Gamma (M,TM)$ .

For concreteness let us work out $T{S}^2$ . Let the pair $U_{\mathrm{N}}\equiv S^2-\{\text{South Pole}\}$ and $U_{\mathrm{S}}\equiv S^2-\{\text{North Pole}\}$ be an open covering of $S^2$ . Let $(X,Y)$ and $(U,V)$ be the respective stereographic coordinates (example 8.1). They are related as

$$U=X/\left(X^2+Y^2\right)V=-Y/\left(X^2+Y^2\right).\text{\hspace{1em}(9.22)}$$

Take $u\in T{S}^2$ such that $\pi (u)=p\in U_{\mathrm{N}}\cap U_{\mathrm{S}}$ . Let ${\varphi }_{\mathrm{N}}$ and ${\varphi }_{\mathrm{S}}$ be the respective local trivializations such that ${\varphi }_{\mathrm{N}}^{-1}(u)=(p,V_{\mathrm{N}}^{\mu })$ and ${\varphi }_{\mathrm{S}}^{-1}(u)=(p,V_{\mathrm{S}}^{\mu })$ . The transition function is

$$t_{\mathrm{SN}}(p)=\frac{\partial (U,V)}{\partial (X,Y)}=\frac{1}{r^2}\left(\begin{array}{cc}-\cos 2\theta & -\sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right)\text{\hspace{1em}(9.23)}$$

where we have put $X=r\cos \theta$ and $Y=r\sin \theta$ . The transition of the components of the tangent vectors consists of a rotation of $\{V_i^{\mu }\}$ by an angle $2\theta$ followed by a rescaling. The reader should verify that $t_{\mathrm{NS}}(p)=t_{\mathrm{SN}}(p{)}^{-1}$ .

Example 9.4

Let $M$ be an $m$ -dimensional manifold embedded in ${\mathbb{R}}^{m+k}$ . Let $N_{p}M$ be the vector space which is normal to $T_{p}M$ in ${\mathbb{R}}^{m+k}$ , that is, $U\cdot V=0$ with respect to the Euclidean metric in ${\mathbb{R}}^{m+k}$ for any $U\in N_{p}M$ and $V\in T_{p}M$ . The vector space $N_{p}M$ is isomorphic to ${\mathbb{R}}^k$ . The normal bundle

$$NM\equiv \bigcup _{p\in M}{N}_{p}M$$

is a vector bundle with the typical fibre ${\mathbb{R}}^k$ .

Consider the sphere $S^2$ embedded in ${\mathbb{R}}^3$ . The normal bundle $N{S}^2$ is imagined as $S^2$ whose surface is pierced perpendicularly by straight lines. $N{S}^2$ is a trivial bundle $S^2\times \mathbb{R}$ .

A vector bundle whose fibre is one-dimensional ( $F=\mathbb{R}$ or $\mathbb{C}$ ) is called a line bundle. A cylinder $S^1\times \mathbb{R}$ is a trivial $\mathbb{R}$ -line bundle. A Möbius strip is also a real line bundle. The structure group $\mathrm{GL}(1,\mathbb{R})=\mathbb{R}-\{0\}$ or $\mathrm{GL}(1,\mathbb{C})=\mathbb{C}-\{0\}$ is Abelian.

In the following, we often consider the canonical line bundle $L$ . Recall that an element $p$ of ${\mathbb{CP}}^n$ is a complex line in ${\mathbb{C}}^{n+1}$ through the origin (example 8.3). The fibre ${\pi }^{-1}(p)$ of $L$ is defined to be the line in ${\mathbb{C}}^{n+1}$ which belongs to $p$ . More formally, let $I^{n+1}\equiv {\mathbb{CP}}^n\times {\mathbb{C}}^{n+1}$ be a trivial bundle over ${\mathbb{CP}}^n$ . If we write an element of $I^{n+1}$ as $(p,v)$ , $p\in {\mathbb{CP}}^n$ , $v\in {\mathbb{C}}^{n+1}$ , $L$ is defined by

$$L\equiv \{(p,v)\in I^{n+1}|v=ap,a\in \mathbb{C}\}.$$

The projection is $(p,v)\stackrel{\pi }{\to }p$

Example 9.5

The (trivial) complex line bundle $L={\mathbb{R}}^3\times \mathbb{C}$ is associated with the non-relativistic quantum mechanics defined on ${\mathbb{R}}^3$ . The wavefunction $\psi (x)$ is simply a section of $L$ .

Let us consider a wavefunction $\psi (x)$ in the field of a magnetic monopole studied in section 1.9. When a monopole is at the origin, $\psi (x)$ is defined on ${\mathbb{R}}^3-\{0\}$ and we have a complex line bundle over ${\mathbb{R}}^3-\{0\}$ . If we are interested only in the wavefunction on $S^2$ surrounding the monopole, we have a complex line bundle over $S^2$ . Note that $S^2$ is a deformation retract of ${\mathbb{R}}^3-\{0\}$ .

9.3.2 Frames

On a tangent bundle $TM$ , each fibre has a natural basis $\{\partial /\partial x^{\mu }\}$ given by the coordinate system $x^{\mu }$ on a chart $U_i$ . We may also employ the orthonormal basis $\{{\hat{e}}_{\alpha }\}$ if $M$ is endowed with a metric. $\partial /\partial x^{\mu }$ or $\{{\hat{e}}_{\alpha }\}$ is a vector field on $U_i$ and the set $\{\partial /\partial x^{\mu }\}$ or $\{{\hat{e}}_{\alpha }\}$ forms linearly independent vector fields over $U_i$ . It is always possible to choose $m$ linearly independent tangent vectors over $U_i$ but it is not necessarily the case throughout $M$ . By definition, the components of the basis vectors are

$$\partial /\partial x^{\mu }=(0,\dots ,0,1,0,\dots ,0)\mu$$

or

$${\hat{e}}_{\alpha }=(0,\dots ,0,1,0,\dots ,0).$$

These vectors define a (local) frame over $U_i$ , see later.

Let $E\stackrel{\pi }{\to }M$ be a vector bundle whose fibre is ${\mathbb{R}}^k$ (or ${\mathbb{C}}^k$ ). On a chart $U_i$ , the piece ${\pi }^{-1}(U_i)$ is trivial, ${\pi }^{-1}(U_i)\cong U_i\times {\mathbb{R}}^k$ , and we may choose $k$ linearly independent sections $\{e_1(p),\dots ,e_k(p)\}$ over $U_i$ . These sections are said to define a frame over $U_i$ . Given a frame over $U_i$ , we have a natural map $F_p\to F$ ( $={\mathbb{R}}^k$ or ${\mathbb{C}}^k$ ) given by

$$V=V^{\alpha }{e}_{\alpha }(p)⟼\left\{V^{\alpha }\right\}\in F.\text{\hspace{1em}(9.24)}$$

The local trivialization is

$${\varphi }_i^{-1}(V)=(p,\{V^{\alpha }(p)\}).\text{\hspace{1em}(9.25)}$$

By definition, we have

$${\varphi }_i\left(p,\{0,\dots ,0,1,0,\dots ,0\}\right)=e_{\alpha }(p).\text{\hspace{1em}(9.26)}$$

Let $U_i\cap U_j\ne \varnothing$ and consider the change of frames. We have a frame $\{e_1(p),\dots ,e_k(p)\}$ on $U_i$ and $\{{\tilde{e}}_1(p),\dots ,{\tilde{e}}_k(p)\}$ on $U_j$ , where $p\in U_i\cap U_j$ . A vector ${\tilde{e}}_{\beta }(p)$ is expressed as

$${\tilde{e}}_{\beta }(p)=e_{\alpha }(p)G(p{)}_{\beta }^{\alpha }\text{\hspace{1em}(9.27)}$$

where $G(p{)}_{\beta }^{\alpha }\in \mathrm{GL}(k,\mathbb{R})$ or $\mathrm{GL}(k,\mathbb{C})$ . Any vector $V\in {\pi }^{-1}(p)$ is expressed as

$$V=V^{\alpha }{e}_{\alpha }(p)={\tilde{V}}^{\alpha }{\tilde{e}}_{\alpha }(p).\text{\hspace{1em}(9.28)}$$

From (9.27) and (9.28) we find that

$${\tilde{V}}^{\beta }=G^{-1}(p{)}_{\alpha }^{\beta }{V}^{\alpha }\text{\hspace{1em}(9.29)}$$

where $G^{-1}(p{)}_{\alpha }^{\beta }G(p{)}_{\gamma }^{\alpha }=G(p{)}_{\alpha }^{\beta }{G}^{-1}(p{)}_{\gamma }^{\alpha }={\delta }_{\gamma }^{\beta }$ . Thus, we find that the transition function $t_{ji}(p)$ is given by a matrix $G^{-1}(p)$ .

9.3.3 Cotangent bundles and dual bundles

The cotangent bundle $T^{\ast }M\equiv {\bigcup }_{p\in M}{T}_p^{\ast }M$ is defined similarly to the tangent bundle. On a chart $U_i$ whose coordinates are $x^{\mu }$ , the basis of $T_p^{\ast }M$ is taken to be $\{\mathrm{d}{x}^1,\dots ,\mathrm{d}{x}^m\}$ , which is dual to $\{\partial /\partial x^{\mu }\}$ . Let $y^{\mu }$ be the coordinates of $U_j$ such that $U_i\cap U_j\ne \varnothing$ . For $p\in U_i\cap U_j$ , we have the transformation,

$$\mathrm{d}{y}^{\mu }=\mathrm{d}{x}^{\nu }{\left(\frac{\partial y^{\mu }}{\partial x^{\nu }}\right)}_p.\text{\hspace{1em}(9.30)}$$

A one-form $\omega$ is expressed, in both coordinate systems, as

$$\omega ={\omega }_{\mu }\mathrm{d}{x}^{\mu }={\tilde{\omega }}_{\mu }\mathrm{d}{y}^{\mu }$$

from which we find that

$${\tilde{\omega }}_{\mu }=G_{\mu }^v(p){\omega }_v\text{\hspace{1em}(9.31)}$$

where $G_{\mu }^{\nu }(p)\equiv (\partial x^{\nu }/\partial y^{\mu }{)}_p$ corresponds to the transition function $t_{ji}(p)$ . Note that $\Gamma (M,T^{\ast }M)={\Omega }^1(M)$ .

This cotangent bundle is easily extended to more general cases. Given a vector bundle $E\stackrel{\pi }{\to }M$ with the fibre $F$ , we may define its dual bundle $E^{\ast }\stackrel{\pi }{\to }M$ . The fibre $F^{\ast }$ of $E^{\ast }$ is the set of linear maps of $F$ to $\mathbb{R}$ (or $\mathbb{C}$ ). Given a general basis $\{e_{\alpha }(p)\}$ of $F_p$ , we define the dual basis $\{{\theta }^{\alpha }(p)\}$ of $F_p^{\ast }$ by $⟨{\theta }^{\alpha }(p),e_{\beta }(p)⟩={\delta }_{\beta }^{\alpha }$ .

9.3.4 Sections of vector bundles

Let $s$ and $s^{\prime }$ be sections of a vector bundle $E\stackrel{\pi }{\to }M$ . The vector addition and the scalar multiplication are pointwisely defined as

$$(s+s^{\prime })(p)=s(p)+s^{\prime }(p)\text{\hspace{1em}(9.32a)}$$

$$(fs)(p)=f(p)s(p)\text{\hspace{1em}(9.32b)}$$

where $p\in M$ and $f\in \mathcal{F}(M)$ . The null vector 0 of each fibre is left invariant under $\mathrm{GL}(k,\mathbb{R})$ (or $\mathrm{GL}(k,\mathbb{C})$ ) and plays a distinguished role. Any vector bundle $E$ admits a global section called the null section $s_0\in \Gamma (M,E)$ such that ${\varphi }_i^{-1}(s_0(p))=(p,0)$ in any local trivialization.

For example, let us consider sections of the canonical line bundle $L$ over ${\mathbb{CP}}^n$ . Let ${\xi }_{(\mu )}^v$ be the inhomogeneous coordinates and $\{z^{\nu }\}$ be the homogeneous coordinates on $U_{\mu }$ . The local section $s_{\mu }$ over $U_{\mu }$ is of the form

$$s_{\mu }=\{{\xi }_{(\mu )}^0,\dots ,1,\dots ,{\xi }_{(\mu )}^n\}\in {\mathbb{C}}^{n+1}.$$

The transition from one coordinate system to the other is carried out by a scalar multiplication: $s_{\nu }=(z^{\mu }/z^{\nu })s_{\mu }$ . Let $L^{\ast }$ be the dual bundle of $L$ . Corresponding to $s_{\mu }$ , we may choose a dual section $s_{\mu }^{\ast }$ such that $s_{\mu }^{\ast }(s_{\mu })=1$ . From this, we find that the transition function of $s_{\mu }^{\ast }$ is a multiplication by $z^{\nu }/z^{\mu }$ , $s_{\nu }^{\ast }=(z^{\nu }/z^{\mu })s_{\mu }^{\ast }$ .

A fibre metric $h_{\mu \nu }(p)$ is also defined pointwisely. Let $s$ and $s^{\prime }$ be sections over $U_i$ . The inner product between $s$ and $s^{\prime }$ at $p$ is defined by

$$(s,s^{\prime }{)}_p=h_{\mu v}(p)s^{\mu }(p)s^{\prime v}(p)\text{\hspace{1em}(9.33a)}$$

if the fibre is ${\mathbb{R}}^k$ . If the fibre is ${\mathbb{C}}^k$ we define

$$(s,s^{\prime }{)}_p=h_{\mu \nu }(p)\overline{s^{\mu }(p)}{s}^{\prime \nu }(p).\text{\hspace{1em}(9.33b)}$$

We have more about this subject in section 10.4.

9.3.5 The product bundle and Whitney sum bundle

Let $E\stackrel{\pi }{\to }M$ and $E^{\prime }\stackrel{{\pi }^{\prime }}{\to }{M}^{\prime }$ be vector bundles with fibres $F$ and $F^{\prime }$ respectively. The product bundle

$$E\times E^{\prime }\stackrel{\pi \times {\pi }^{\prime }}{\to }M\times M^{\prime }\text{\hspace{1em}(9.34)}$$

is a fibre bundle whose typical fibre is $F\oplus F^{\prime }$ . [A vector in $F\oplus F^{\prime }$ is written as

$$\left(\begin{array}{c}V\\ W\end{array}\right)\text{w h e r e}V\in F\text{a n d}W\in F^{\prime }.$$

Vector addition and scalar multiplication are defined by

$$\left(\begin{array}{c}V\\ W\end{array}\right)+\left(\begin{array}{c}{V}^{\prime }\\ W^{\prime }\end{array}\right)=\left(\begin{array}{c}V+V^{\prime }\\ W+W^{\prime }\end{array}\right)$$

and

$$\lambda \left(\begin{array}{c}V\\ W\end{array}\right)=\left(\begin{array}{c}\lambda V\\ \lambda W\end{array}\right).$$

Let $\{e_{\alpha }\}$ and $\{f_{\beta }\}$ be bases of $F$ and $F^{\prime }$ respectively. Then $\{e_{\alpha }\}\cup \{f_{\beta }\}$ is a basis of $F\oplus F^{\prime }$ and we find that $\dim (F\oplus F^{\prime })=\dim F+\dim F^{\prime }$ .] If $\pi (u)=p$ and ${\pi }^{\prime }(u^{\prime })=p^{\prime }$ the projection $\pi \times {\pi }^{\prime }$ acts on $(u,u^{\prime })\in E\times E^{\prime }$ as

$$\pi \times {\pi }^{\prime }(u,u^{\prime })=(p,p^{\prime }).\text{\hspace{1em}(9.35)}$$

The fibre at $(p,p^{\prime })$ is $F_p\oplus F_{p^{\prime }}^{\prime }$ . For example, if $M=M_1\times M_2$ , we have $TM=T{M}_1\times T{M}_2$ .

Let $E\stackrel{\pi }{\to }M$ and $E^{\prime }\stackrel{{\pi }^{\prime }}{\to }M$ be vector bundles with fibres $F$ and $F^{\prime }$ respectively. The Whitney sum bundle $E\oplus E^{\prime }$ is a pullback bundle of $E\times E^{\prime }$ by $f:M\to M\times M$ defined by $f(p)=(p,p)$ ,

$$\begin{array}{rl}& \begin{array}{ccc}E\oplus E^{\prime }& \stackrel{{\pi }_2}{\to }& E\times E^{\prime }\\ {\pi }_1\downarrow & & \downarrow \pi \times {\pi }^{\prime }\end{array}\\ & M\stackrel{f}{⟶}M\times M.\end{array}\text{\hspace{1em}(9.36)}$$

Thus, $E\oplus E^{\prime }=\{(u,u^{\prime })\in E\times E^{\prime }|\pi \times {\pi }^{\prime }(u,u^{\prime })=(p,p)\}$ . The fibre of a Whitney sum bundle is $F\oplus F^{\prime }$ . $(\pi \times {\pi }^{\prime }{)}^{-1}(p)$ is isomorphic to ${\pi }^{-1}(p)\oplus {\pi }^{\prime -1}(p)=F_p\oplus F_p^{\prime }$ . In short, $E\oplus E^{\prime }$ is a bundle over $M$ whose fibre at $p$ is $F_p\oplus F_p^{\prime }$ . Let $\{U_i\}$ be an open covering of $M$ and $\{t_{ij}^E\}$ and $\{t_{ij}^{\prime }\}$ be the transition functions of $E$ and $E^{\prime }$ respectively. Then the transition function $T_{ij}$ of $E\oplus E^{\prime }$ is a $(\dim F+\dim F^{\prime })\times (\dim F+\dim F^{\prime })$ matrix

$$T_{ij}(p)=\left(\begin{array}{cc}{t}_{ij}^E(p)& 0\\ 0& t_{ij}^{E^{\prime }}(p)\end{array}\right)\text{\hspace{1em}(9.37)}$$

which acts on $F\oplus F^{\prime }$ on the left.

Example 9.6

Let $E=T{S}^2$ and $E^{\prime }=N{S}^2$ defined in ${\mathbb{R}}^3$ . Take $u\in T{S}^2$ and $v\in N{S}^2$ whose local trivializations are ${\varphi }_i^{-1}(u)=(p,V)$ and ${\psi }_i^{-1}(v)=(q,W)$ , respectively, where $p,q\in S^2$ , $V\in {\mathbb{R}}^2$ and $W\in \mathbb{R}$ . If $(u,v)$ is a point of the product bundle $E\times E^{\prime }$ , we have a trivialization ${\Phi }_{i,j}={\varphi }_i\times {\psi }_j$ such that

$${\Phi }_{i,j}^{-1}(u,v)=(p,q;V,W).\text{\hspace{1em}(9.38a)}$$

If, however, $(u,v)\in E\oplus E^{\prime }$ , $u$ and $v$ satisfy the stronger condition $\pi (u)={\pi }^{\prime }(v)$ $(=p$ , say). Thus, we have

$${\Phi }_i^{-1}(u,v)=(p;V,W).\text{\hspace{1em}(9.38b)}$$

The Whitney sum $T{S}^2\oplus N{S}^2$ , $S^2$ being embedded in ${\mathbb{R}}^3$ , is a trivial bundle over $S^2$ , whose fibre is isomorphic to ${\mathbb{R}}^3$ .

and

9.3.6 Tensor product bundles

Let $E\stackrel{\pi }{\to }M$ and $E^{\prime }\stackrel{{\pi }^{\prime }}{\to }M$ be vector bundles over $M$ . The tensor product bundle $E\otimes E^{\prime }$ is obtained by assigning the tensor product of fibres $F_p\otimes F_p^{\prime }$ to each point $p\in M$ . If $\{e_{\alpha }\}$ and $\{f_{\beta }\}$ are bases of $F$ and $F^{\prime }$ , $F\otimes F^{\prime }$ is spanned by $\{e_{\alpha }\otimes f_{\beta }\}$ and, hence, $\dim (E\otimes E^{\prime })=\dim E\times \dim E^{\prime }$ .

Let ${⨂}^{r}E\equiv E\otimes \cdots \otimes E$ be the tensor product bundle of $rE$ . If $\{e_{\alpha }\}$ is the basis of the fibre $F$ of $E$ , the fibre of ${⨂}^{r}E$ is spanned by $\{e_{{\alpha }_1}\otimes \cdots \otimes e_{{\alpha }_r}\}$ . If we define $\wedge$ by

$$e_{\alpha }\wedge e_{\beta }\equiv e_{\alpha }\otimes e_{\beta }-e_{\beta }\otimes e_{\alpha }\text{\hspace{1em}(9.39)}$$

we have a bundle ${\wedge }^r(E)$ of totally anti-symmetric tensors spanned by $\{e_{{\alpha }_1}\wedge$ $\cdots \wedge e_{{\alpha }_r}\}$ . In particular, ${\Omega }^r(M)$ , the space of $r$ -forms on $M$ , is identified with $\Gamma (M,{\Lambda }^r(T^{\ast }M))$ .

Exercise 9.1

Let $E_1$ , $E_2$ and $E_3$ be vector bundles over $M$ . Show that $\otimes$ is distributive:

$$E_1\otimes (E_2\oplus E_3)=(E_1\otimes E_2)\oplus (E_1\otimes E_3).\text{\hspace{1em}(9.40)}$$

Express the transition functions of $E_1\otimes (E_2\oplus F_3)$ in terms of those of $E_1,E_2$ and $E_3$ .

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