[1] 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.
[1] If the bundle π: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.
where V is a vector space transforming covariantly under some representation of G. If the base space M is also a metric space, with metric g, and the principal bundle is endowed with a connection form ω, then π*g+kω is a metric defined on the entire tangent bundle E = TP.
This bundle metric underpins the generalized form of Kaluza–Klein theory due to several interesting properties that it possesses.
The scalar curvature derived from this metric is constant on each fiber,[2] this follows from the Ad(G) invariance of the fiber metric k. The scalar curvature on the bundle can be decomposed into three distinct pieces: where RE is the scalar curvature on the bundle as a whole (obtained from the metric π*g+kω above), and RM(g) is the scalar curvature on the base manifold M (the Lagrangian density of the Einstein–Hilbert action), and L(g, ω) is the Lagrangian density for the Yang–Mills action, and RG(k) is the scalar curvature on each fibre (obtained from the fiber metric k, and constant, due to the Ad(G)-invariance of the metric k).