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 pi : E → M a vector bundle on M, then a metric on E is a bundle map k : 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 k to each fibre over M is a nondegenerate bilinear map of vector spaces. Roughly speaking, k gives a kind of dot product (not necessarily symmetric or positive definite) on the vector space above each point of M, and these products vary smoothly over M. Every vector bundle with paracompact base space can be equipped with a bundle metric. For a vector bundle of rank n, this follows from the bundle charts : the bundle metric can be taken as the pullback of the inner product of a metric on ; for example, the orthonormal charts of Euclidean space. The structure group of such a metric is the orthogonal group O(n). If M is a Riemannian manifold, and E is its tangent bundle TM, then the Riemannian metric gives a bundle metric, and vice versa. If the bundle pi:P → M is a principal fiber bundle with group G, and G is a compact Lie group, then there exists an Ad(G)-invariant inner product k on the fibers, taken from the inner product on the corresponding compact Lie algebra. More precisely, there is a metric tensor k defined on the vertical bundle E = VP such that k is invariant under left-multiplication: for vertical vectors X, Y and Lg is left-multiplication by g along the fiber, and Lg* is the pushforward. That is, E is the vector bundle that consists of the vertical subspace of the tangent of the principal bundle. More generally, whenever one has a compact group with Haar measure μ, and an arbitrary inner product h(X,Y) defined at the tangent space of some point in G, one can define an invariant metric simply by averaging over the entire group, i.e. by defining as the average.