A canonical connection associated with certain G -structures.

Abstract

Let P be a G-structure on a manifold M and AdP be the adjoint bundle of P. The authors deduce the following main result: there exists a unique connection r adapted to P such that trace(S iX Tor(r)) = 0 for every section S of AdP and every vector field X on M, provided Tor(r) stands for the torsion tensor field of r. Two examples, namely almost Hermitian structures and almost contact metric structures, are discussed in more detail. Another interesting result reads: for a given structure group G, if it is possible to attach a connection to each G-structure in a functorial way with the additional assumption that the connection depends on first order contact only, then the first prolongation of the Lie algebra of G vanishes