As is well known, any G-invariant metric g assigned on a pricipal G-bundle P is "dimensionally reducible", i.e., it uniquely defines in a canonical way a principal connection, the latter being the unique "horizontal" distribution which is g-orthogonal to the distribution of all vertical subspaces . We generalize the previous idea to the affine framework, showing the existence of a constructive standard procedure which uniquely generates a principal connection from any given linear connection belonging to a very large subclass of G-invariant linear connections in P, which we call "dimensionally reducible linear connections". These are in fact a dense open subset of all "totally vertical connections", i.e., connections such that the covariant derivatives along vertical vectorfields transform vertical vectors into vertical vectors. The main result is the following: under the only hypothesis that the structure group G of the bundle is a reductive subgroup of some linear group Gl(n,I\d\ba1()R), any "totally vertical" linear connection which satisfies a mild regularity condition is "dimensionally reducible". This result finds interesting applications in the affine formulation of so-called "Kaluza-Klein theories".
Reducibility of G-invariant linear connections in principal G-bundles
FERRARIS, Marco;FRANCAVIGLIA, Mauro;
1992-01-01
Abstract
As is well known, any G-invariant metric g assigned on a pricipal G-bundle P is "dimensionally reducible", i.e., it uniquely defines in a canonical way a principal connection, the latter being the unique "horizontal" distribution which is g-orthogonal to the distribution of all vertical subspaces . We generalize the previous idea to the affine framework, showing the existence of a constructive standard procedure which uniquely generates a principal connection from any given linear connection belonging to a very large subclass of G-invariant linear connections in P, which we call "dimensionally reducible linear connections". These are in fact a dense open subset of all "totally vertical connections", i.e., connections such that the covariant derivatives along vertical vectorfields transform vertical vectors into vertical vectors. The main result is the following: under the only hypothesis that the structure group G of the bundle is a reductive subgroup of some linear group Gl(n,I\d\ba1()R), any "totally vertical" linear connection which satisfies a mild regularity condition is "dimensionally reducible". This result finds interesting applications in the affine formulation of so-called "Kaluza-Klein theories".I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.