Bundle metric

Let $\pi:E\to M$ be a smooth real vector bundle over a smooth manifold . A bundle metric on $E$ is an assignment of an inner product

for each $x\in M$ such that:

• Each $\langle\cdot,\cdot\rangle_x$ is a positive-definite symmetric bilinear form on the real vector space $E_x$.
• For any smooth local sections $s,t:U\to E$ (defined on an open set $U\subseteq M$), the function $U\to\mathbb R,\qquad x\mapsto \langle s(x),t(x)\rangle_x$ is smooth.

Equivalently, in any local frame $(e_1,\dots,e_r)$ over $U$, the matrix-valued function $G=(G_{ij})$ with

is a smooth map $U\to \mathrm{Sym}^+(r,\mathbb R)$ (positive-definite symmetric matrices), and transforms under changes of frame by the usual congruence rule.

A bundle metric is the same as a reduction of the frame bundle from the general linear group to the orthogonal group, yielding the orthonormal frame bundle .

Examples

1. Riemannian metric. A Riemannian metric on $M$ is precisely a bundle metric on the tangent bundle $TM$.

2. Standard metric on a trivial bundle. On $E=M\times\mathbb R^r$, the standard Euclidean inner product on $\mathbb R^r$ defines a bundle metric with constant matrix $I_r$ in the standard frame.

3. Induced metrics. A bundle metric on $E$ induces canonical bundle metrics on $E^*$, on $E\oplus F$, and on tensor and exterior powers (e.g. on $\Lambda^kE$) by the standard linear-algebraic constructions performed fiberwise.