  Many features of dimensional reduction schemes are determined by the breaking
of higher dimensional general covariance associated with the selection of a
particular subset of coordinates. By investigating residual covariance we
introduce lower dimensional tensors --generalizing to one side Kaluza-Klein
gauge fields and to the other side extrinsic curvature and torsion of embedded
spaces-- fully characterizing the geometry of dimensional reduction. We obtain
general formulas for the reduction of the main tensors and operators of
Riemannian geometry. In particular, we provide what is probably the maximal
possible generalization of Gauss, Codazzi and Ricci equations and various other
standard formulas in Kaluza-Klein and embedded spacetimes theories. After
general covariance breaking, part of the residual covariance is perceived by
effective lower dimensional observers as an infinite dimensional gauge group.
This reduces to finite dimensions in Kaluza-Klein and other few remarkable
backgrounds, all characterized by the vanishing of appropriate lower
dimensional tensors.
