Mathematical aspects of Kaluza-Klein gravity

We give a coordinate-free calculation of the Ricci tensor (Ric) over bar and Ricci scalar S for a general Kaluza-Klein metric (g) over bar on a fiber bundle pi : E --> M over a semi-Riemannian manifold (M, g). The metric (g) over bar is built from the spacetime metric g, a connection sigma : E x TM --> TE, and a fiber metric h on the vertical bundle (or internal space) VE of TE. The resulting formulas for (Ric) over bar and S are shown to involve new geometric objects: the gauge Hessian H-h(sigma) and gauge Laplacian Delta(sigma)h, as well as other globally defined quantities. These formulas appear to be the first global version of the many local coordinate versions existing in the literature. Additionally we isolate a class of fiber metrics h and connections sigma for which these formulas reduce considerably in complexity. The higher-dimensional field equations, (Ric) over bar - (1 /2)(S) over tilde(g) over bar = (1/2)A (g) over bar + 8pi(T) over bar, contain the field equations for gravity and gauge fields, but generally the fields depend on the fiber coordinates. However, this dependence can be eliminated if one restricts attention to principal bundles with equivariant fiber metrics and connections. (C) 2003 Elsevier B.V. All rights reserved.