In differential geometry, the notion of a metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 can be extended to an arbitrary vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

. Specifically, if M is a topological manifold

Topological manifold

In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...

 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

Bundle map

In mathematics, a bundle map is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely...

 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.