[Tu Differential Geometry] 10. Connections on a Vector Bundle

 This article is one of Manifold, Differential Geometry, Fibre Bundle.

10.1 Connections on a Vector Bundle

Definition 10.1 Let $E\rightarrow M$ be a $C^\infty$ vector bundle over $M$. A connection on $E$ is a map $$\nabla:\mathfrak{X}(M)\times \Gamma(E)\rightarrow \Gamma(E)$$ such that for $X\in \mathfrak{X}(M)$ and $s\in \Gamma(E)$,

(i) $\nabla_Xs$ is $\mathcal{F}$-linear in $X$ and $\mathbb{R}$-linear in $s$;

(ii) (Leibniz rule) if $f$ is a $C^\infty$ function on $M$, then $$\nabla_X(fs)=(Xf)s+f\nabla_Xs.$$

Since $Xf=(df)(X)$, the Leibniz rule may be written as $$\nabla_X(fs)=(df)(X)s+f\nabla_Xs,$$ or, suppressing $X$, $$\nabla(fs)=df\cdot s+f\nabla s.$$

We say that a section $s\in \Gamma(E)$ is flat if $\nabla_Xs=0$ for all $X\in \mathfrak{X}(M)$.

Example 10.3 (Induced connection on a trivial bundle). Let $E$ be a trivial bundle of rank $r$ over a manifold $M$. Thus, there is a bundle isomorphism $\phi:E\rightarrow M\times \mathbb{R}^r$, called a trivialization for $E$ over $M$. The trivialization $\phi:E\rightarrow M\times \mathbb{R}^r$ induces a connection on $E$ as follows. If $v_1,\cdots,v_r$ is a basis for $\mathbb{R}^r$, then $s_i:p\mapsto (p,v_i)$, $i=1,\cdots,r$ define a global frame for the product bundle $M\times \mathbb{R}^r$ over $M$, and $e_i=\phi^{-1}\circ s_i$, $i=1,\cdots,r$, define a global frame for $E$ over $M$: 

So every section $s\in Gamma(E)$ can be written uniquely as a linear combination $$s=\sum h^ie_i,\quad h^i\in \mathcal{F}.$$

We can define a connection $\nabla$ on $E$ by declaring the sections $e_i$ to be flat and applying the Leibniz rule and $\mathbb{R}$-linearity to define $\nabla_Xs$: $$\begin{align}\nabla_Xs=\nabla_X\left( \sum h^ie_i\right) =\sum (Xh^i)e_i.\end{align}$$

10.2 Exitence of a Connection on a Vector Bundle

Let $\nabla^0$ and $\nabla^1$ be two connections on a vector bundle $E$ over $M$. By the Leibniz rule, for any vector field $X\in \mathfrak{X}(M)$, section $s\in \Gamma(E)$, and function $f\in C^\infty(M)$, $$\begin{align}\nabla^0_X(fs)=(Xf)s+f\nabla_X^0s,\\ \nabla^1_X(fs)=(Xf)s+f\nabla_X^1s.\end{align}$$ Hence, $$\begin{align} (\nabla^0_X+\nabla^1_X)(fs)=2(Xf)s+f(\nabla_X^0+\nabla_X^1)s.\end{align}$$ Because of the extra factor of 2 in (4) the sum of two connections does not satisfy the Leibniz rule and so is not a connection. However, if we multiply (2) by $1-t$ and (3) by $t$, then $(1-t)\nabla_X^0+t\nabla_X^1$ satisfies the Leibniz rule. More generally, the same idea can be use to prove the following proposition.

Proposition 10.5. Any finite linear combination $\sum t_i \nabla^i$ of connections $\nabla^i$ is a connection provided the coefficients add up to $1$, that is, $\sum t_i=1$.

A finite linear combination whose coefficients add up to $1$ is called a convex linear combination. Using Proposition 10.5, we not prove the existence of a connection.

Theorem 10.6 (Exitence of Connection on smooth vector bundle). Every $C^\infty$ vector bundle $E$ over a manifold $M$ has a connection.

Proof. Fix a trivializing open cover $\{U_\alpha\}$ for $E$ and a partition of unity $\{\rho_\alpha\}$ subordiante to $\{U_\alpha\}$ ($\sum \rho_\alpha=1$). On each $U_\alpha$, the vector bundle $E|_{U_\alpha}$ is trivial and so has a connection $\nabla^\alpha$ by Example 10.3. For $X\in \mathfrak{X}(M)$ and $s\in \Gamma(E)$, denote by $s_\alpha$ the restriction of $s$ to $U_\alpha$ and define $$\nabla_Xs=\sum \rho_\alpha \nabla_X^\alpha s_\alpha.$$

By checking on the open set $U$, it is easy to show that $\nabla_Xs$ is $\mathcal{F}$-linear in $X$, is $\mathbb{R}$-linear in $s$, and satisfies the Leibniz rule. Hence, $\nabla$ is a connection on $E$.

10.3 Curvature of a Connection on a Vector Bundle

The concept of torsion does not make sense for the connection on an arbitrary vector bundle, but curvature still odes. It is defined by the same formula as for an affine connection: for $X,Y\in\mathfrak{X}(M)$ and $s\in \Gamma(E)$, $$R(X,Y)s=\nabla_X \nabla_Ys-\nabla_Y\nabla_Xs-\nabla_{[X,Y]}s\in \Gamma(E).$$ So $R$ is an $\mathbb{R}$-multilinear map $$\mathfrak{X}(M)\times \mathfrak{X}(M)\times \Gamma(E)\rightarrow \Gamma(E).$$ As before, $R(X,Y)s$ is $\mathcal{F}$-linear in all three arguments and so it is actually defined pointwise. Moreover, because $R_p(X_p,Y_p)$ is skew-symmetric in $X_p$ and $Y_p$, at every point $p$ there is an alternating bilinear map $$R_p:T_pM\times T_pM\rightarrow \mbox{Hom}(E_p,E_p)=:\mbox{End}(E_p)$$ into the endomorphism ring of $E_p$. We call this map the curvature tensor of the connection $\nabla$.

10.4 Riemannian Bundles

We can also generalize the notion of a Riemannian metric to vector bundles. Let $E\rightarrow M$ be a $C^\infty$ vector bundle over a manifold $M$. A Riemannian metric on $E$ assigns to each $p\in M$ and inner product $\left\langle\ ,\ \right\rangle_p$ on the fiber $E_p$; the assignment is required to be $C^\infty$ in the following sense: if $s$ and $t$ are $C^\infty$ sections of $E$, then $\left\langle s,t\right\rangle$ is a $C^\infty$ function on $M$.

Thus, a Riemannian metric on a manifold $M$ is simply a Riemannian metric on the tangent bundle $TM$. A vector bundle together with a Riemannian metric is called a Riemannian bundle.

Example 10.7 (Riemannian bundle of trivial vector bundle). Let $E$ be a trivial vector bundle of rank $r$ over a manifold $M$, with trivialization $\phi:E\rightarrow M\times \mathbb{R}^r$. The Euclidean inner product $\left\langle\ ,\ \right\rangle_{\mathbb{R}^r}$ on $\mathbb{R}^r$ induces a Riemannian metric on $E$ via the trivialization $\phi:$ if $u,v\in E_p$, then the fiber map $\phi_p:E_p\rightarrow \mathbb{R}^r$ is a linear isomorphism and we define $$\left\langle u,v\right\rangle = \left\langle \phi_p(u),\phi_p(v)\right\rangle_{\mathbb{R}^r}.$$ It is easy to check that $\left\langle\ ,\ \right\rangle$ is a Riemannian metric on $E$.

Theorem 10.8 (Riemannian bundle of smooth vector bundle). On any $C^\infty$ vector bundle $\pi:E\rightarrow M$, there is a Riemannian metric.

Proof. Let $\{U_\alpha,\ \varphi_\alpha:E|_{U_\alpha}\rightarrow U_\alpha\times \mathbb{R}^k\}$ be a trivializing open cover for $E$. On $E|_{U_\alpha}$ there is a Riemannian metric $\left\langle\ ,\ \right\rangle_\alpha$ induced from the trivial bundle $U_\alpha\times \mathbb{R}^k$. Let $\{\rho_\alpha\}$ be a partition of unity on $M$ subordinate to the open cover $\{U_\alpha\}$. By the same reasoning as in Theorem 1.12, the sum $$\left\langle\ ,\ \right\rangle:=\sum \rho_\alpha \left\langle\ ,\ \right\rangle_\alpha$$ is a finite sum in a neighborhood of each point of $M$ and is a Riemannian metric on $E$.

10.5 Metric Connection

We say that a connection $\nabla$ on a Riemannian bundle $E$ is compatible with the metric if for all $X\in\mathfrak{X}(M)$ and $s,t\in \Gamma(E)$, $$X\left\langle s,t\right\rangle=\left\langle \nabla_Xs,t\right\rangle + \left\langle s,\nabla_Xt\right\rangle.$$ A connection compatible with the metric on a Riemannian bundle is also called a metric connection.

Proposition 10.9 (Compatible of trivial vector bundle). On a tricial vector bundle $E$ over a manifold $M$ with trivialization $\phi:E\rightarrow M\times \mathbb{R}^r$, the connection $\nabla$ on $E$ induced by the trivialization $\phi$ is compatible with the Riemannian metric $\left\langle\ ,\ \right\rangle$ on $E$ induced by the trivialization.

Proof. Let $v_1,\cdots,v_r$ be an orthonormal basis for $\mathbb{R}^r$ and $e_1,\cdots,e_r$ the corresponding global frame for $E$, where $e_i(p)=\phi^{-1}(p,v_i)$. Then $e_1,\cdots,e_r$ is an orthonormal flat frame for $E$ with respect to the Riemannian metric and the connection on $E$ induced by $\phi$. If $s=\sum a^i e_i$ and $t=\sum b^je_j$ are $C^\infty$ sections of $E$, then $$X\left\langle s,t\right\rangle = X\left( \sum a^ib^i\right) =\sum (Xa^i)b^i+\sum a^iXb^i=\left\langle \nabla_Xs,t\right\rangle + \left\langle s,\nabla_Xt\right\rangle$$ (Use orthonormality of $e_1,\cdots,e_r$ and (1).)

Lemma 10.11 (Linear sum of compatible bundles is compatible). Let $E\rightarrow M$ be a Riemannian bundle. Suppose $\nabla^1,\cdots,\nabla^m$ are connections on $E$ compatible with the metric and $a_1,\cdots,a_m$ are $C^\infty$ functions on $M$ that add up to $1$. Then $\nabla=\sum_ia_i\nabla^i$ is a connection on $E$ compatible with the metric.

Proof. By Proposition 10.5, $\nabla$ is a connection on $E$. It remains to check that $\nabla$ is compatible with the metric. If $X\in \mathfrak{X}(M)$ and $s,t\in \Gamma(E)$, then $$\begin{align}X\left\langle s,t\right\rangle = \left\langle \nabla^i_Xs,t\right\rangle + \left\langle s,\nabla^i_Xt\right\rangle\end{align}$$ for all $i$ because $\nabla^i$ is compatible with the metric. Now multiply (5) by $a_i$ and sum: $$X\left\langle s,t\right\rangle = \sum a_iX\left\langle s,t\right\rangle = \left\langle \sum a_i\nabla^i_Xs,t\right\rangle + \left\langle s,\sum a_i\nabla^i_Xt\right\rangle = \left\langle \nabla_Xs,t\right\rangle + \left\langle s,\nabla_Xt\right\rangle.$$

Proposition 10.12 (Exitence of compatible bundle). On any Riemannian bundle $E\rightarrow M$, there is a connection compatible with the metric.

Proof. Choose a trivializing open cover $\{U_\alpha\}$ for $E$. By Proposition 10.9, for each trivializing open set $U_\alpha$ of the cover, we can find a connection $\nabla^\alpha$ on $E|_{U_\alpha}$ compatible with the metric. Let $\{\rho_\alpha\}$ be a partition of unity subordinate to the open cover $\{U_\alpha\}$. Because the collection $\{\mbox{supp}\rho_\alpha\}$ is locally finite, the sum $\nabla=\sum \rho_\alpha \nabla^\alpha$ is a finite sum in a neighborhood of each point. Since $\sum \rho_\alpha\equiv 1$, by Lemma 10.11, $\nabla$ is a connection on $E$ compatible with the metric.

10.6 Restricting a Connection to an Open Subset

A connection $\nabla$ on a vector bundle $E$ over $M$, $$\nabla:\mathfrak{X}(M)\times \Gamma(E)\rightarrow \Gamma(E)$$ is $\mathcal{F}$-linear in the first argument, but not $\mathcal{F}$-linear in the second argument. However, it turns out that the $\mathcal{F}$-linearity in the first argument and the Leibniz rule in the second argument are enough to imply that $\nabla$ is a local operator.

Proposition 10.13. Let $\nabla$ be a connection on a vector bundle $E$ over a manifold $M$, $X$ be a smooth vector field on $M$, and $s$ a smooth section of $E$. If either $X$ or $s$ vanishes identically on an open subset $U$, then $\nabla_Xs$ vanishes identically on $U$.

Proof. Suppose $X$ vanishes identically on $U$. Since $\nabla_Xs$ is $\mathbb{F}$-linear in $X$, by Proposition 7.17, for any $s\in \Gamma(E)$, $\nabla_Xs$ vanishes identically on $U$.

Next, suppose $s\equiv 0$ on $U$ and $p\in U$. Choose a $C^\infty$ bump function $f$ such that $f\equiv 1$ in a neighborhood of $p$ and $\mbox{supp} f\subset U$. By our choice of $f$, the derivative $X_pf$ is zero. Since $s$ vanishes on the support of $f$, we have $fs\equiv 0$ on $M$ and so $\nabla_X(fs)\equiv0$. Evaluating at $p$ gives $$0=(\nabla_X(fs))_p=(Xf)_ps_p+f(p)(\nabla_Xs)_p=(\nabla_Xs)_p.$$ Because $p$ is arbitrary point of $U$, we conclude that $\nabla_Xs$ vanishes identically on $U$.

In the same way that a local operator: $\Gamma(E)\rightarrow \Gamma(F)$ can be ristricted to any open subset (Theroem 7.20), a connection on a vector bundle can be restricted to any open subset. More precisely, given a connection $\nabla$ on a vector bundle $E$, for every open set $U$ there is a connection $$\nabla^U:\mathfrak{X}(U)\times \Gamma(U,E)\rightarrow \Gamma(U,E)$$ such that for any global vector field $\bar{X}\in \mathfrak{X}(M)$ and global section $\bar{s}\in \Gamma(E)$, $$\nabla^U_{\bar{X}|_U}(\bar{s}|_U)=(\nabla_\bar{X}\bar{s})|_U.$$

Suppose $X\in \mathfrak{X}(U)$ and $s\in \Gamma(U,E)$. For any $p\in U$, to define $\nabla_X^Us\in \Gamma(U,E)$ first pick a global vector field $\bar{X}$ and a global section $\bar{s}\in \Gamma(E)$ that agree with $X$ and $s$ in a neighborhood of $p$. Then define $$(\nabla_X^Us)(p)=(\nabla_\bar{X}\bar{s})(p).$$ Because $\nabla_\bar{X}\bar{s}$ is a local operator in $\bar{X}$ and in $\bar{s}$, this definition is independent of the choice of $\bar{X}$ and $\bar{s}$. It is a routine matter to show that $\nabla^U$ satisfies all the properties of a connection on $E|_U$.

10.7 Connections at a Point

Suppose $\nabla$ is a connection on a vector bundle $E$ over a manifold $M$. For $X\in \mathfrak{X}(M)$ and $s\in \Gamma(E)$, since $\nabla_Xs$ is $\mathcal{F}$-linear in $X$, it is a point operator in $X$ and Proposition 7.25 assures us that it can be defined pointwise in $X$: there is a unique map, also denoted by $\nabla$, $$\nabla:T_pM\times \Gamma(E)\rightarrow E_p$$ such that if $X\in \mathfrak{X}(M)$ and $s\in \Gamma(E)$, then $$\nabla_{X_p}s=(\nabla_Xs)_p.$$

It is easy to check that $\nabla_{X_p}s$ has the following properties: for $X_p\in T_pM$ and $s\in \Gamma(E)$, 

(i) $\nabla_{X_p}s$ is $\mathbb{R}$-linear in $X_p$ and in $s$;

(ii) if $f$ is a $C^\infty$ function on $M$, then $$\nabla_{X_p}(fs)=(X_pf)s(p)+f(p)\nabla_{X_p}s.$$