metric (vector bundle)

In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle. Specifically, if M is a topological manifold and E → M a vector bundle on M, then a metric (sometimes called a bundle metric, or fibre metric) on E is a bundle map g : E ×_M E → M × R from the fiber product of E with itself to the trivial bundle with fiber R such that the restriction of g to each fibre over M is a nondegenerate bilinear map of vector spaces.