Reduction of a structure group.

Let $X$ be a smooth manifold and $\pi:E\rightarrow X$ a vector bundle of rank $k$ on X. If one manages to redefine $E$ by using a cocycle $\{g_{\alpha,\beta}\}$ whose values are all contained in a subgroup $G$ of $GL(k)$, then one says that $E$ is a $G$-bundle.

Many textbooks say that $E$ comes with a metric (a smoothly-varying fiber-wise metric), the structure group can be reduced to $O(k)$. I am not really convinced with this fact; starting with a transition function $\{g_{\alpha,\beta}\}$ whose values are just in $GL(k)$, is it obvious that one can make a new transition function whose values are just in $O(k)$?

Solution 1:

If $\{U_{\gamma}\}$ is an open cover of $X$ which trivialises $E$, then over $U_{\alpha}$, there are sections $s_1, \dots, s_m$ such that $\{s_i(x)\ |\ i = 1, \dots, m\}$ is a basis for $E_{x}$ for every $x \in U_{\alpha}$ ($m$ is the rank of $E$). Suppose now that $x \in U_{\alpha}\cap U_{\beta}$, and let $\sigma_1, \dots, \sigma_m$ be the corresponding sections; in particular $\{\sigma_i(x)\ |\ i = 1, \dots, m\}$ is also a basis for $E_x$. Then there is a change of basis matrix $g_{\alpha\beta}(x)$ which transforms $\{\sigma_i(x)\ |\ i = 1, \dots, m\}$ into $\{s_i(x)\ |\ i = 1, \dots, m\}$; that is $s_i(x) = g_{\alpha\beta}(x)\sigma_i(x)$ for $i = 1, \dots, m$.

A choice of metric on $E$ gives an inner product $\langle \cdot, \cdot\rangle_x$ on $E_x$. This allows us to measure angles; in particular, we can apply the Gram-Schmidt process to obtain an orthonormal basis for $E_x$. Therefore, we can take both $\{\sigma_i(x)\ |\ i = 1, \dots, m\}$ and $\{s_i(x)\ |\ i = 1, \dots, m\}$ to be orthonormal bases for $E_x$. Note that

$$\delta_{ij} = \langle s_i(x), s_j(x)\rangle_x = \langle g_{\alpha\beta}(x)\sigma_i(x), g_{\alpha\beta}(x)\sigma_j(x)\rangle_x,$$

and

$$\delta_{ij} = \langle\sigma_i(x), \sigma_j(x)\rangle_x.$$

Therefore $\langle\sigma_i(x), \sigma_j(x)\rangle_x = \langle g_{\alpha\beta}(x)\sigma_i(x), g_{\alpha\beta}(x)\sigma_j(x)\rangle_x$, so $g_{\alpha\beta}(x)$ preserves the inner product and is therefore an element of the orthogonal group.