Bundle metric

In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric.

If M is a topological manifold and π:E → M a vector bundle on M, then a metric on E is a bundle map k : E ×_ME → M × R from the fiber product of E with itself to the trivial bundle with fiber R such that the restriction of k to each fibre over M is a nondegenerate bilinear map of vector spaces.

Every vector bundle can be equipped with a bundle metric. For a vector bundle of rank n, this follows from the bundle charts ${\displaystyle \phi :\pi ^{-1}(U)\to \mathbb {R} ^{n}}$: the bundle metric can be taken as the pullback of the inner product of a metric on ${\displaystyle \mathbb {R} ^{n}}$; for example, the orthonormal charts of Euclidean space. The structure group of such a metric is the orthogonal group O(n).

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