Bundle metric - Imported Knowlpedia Content

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

\langle\cdot,\cdot\rangle_x : E_x\times E_x\to \mathbb R

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.

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.

Equivalent characterizations

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

G_{ij}(x)=\langle e_i(x),e_j(x)\rangle_x

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.

Examples

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

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.

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.