$G$-structures of order two and transgression operators

Abstract

This paper is an addendum to a recent paper by Chern and Simons [1] on curvature forms and conformal transformations.Our purpose is to give a Gstructure theoretic interpretation to their results and to obtain similar results for projective and other second order G-structures in a unified manner.Let P be a differentiate principal bundle over M with group G. Let ω be a connection form and Ω its curvature form on P. Let P(G) denote the set of G-invariant symmetric multilinear forms of degree k on the Lie algebra g of G.If / € P(G), then /(β,, Ω) can be pulled down to a closed 2&-form on the base M to give an element of H 2k (M;R), called the characteristic class defined by /.This class is transgressive.In fact, if we setThe problem we discuss here is to find out how Tf(ώ) depends on the connection ω.Let ω(s), 0 < s < 1, be a 1-parameter family of connections in P and Ω(s) the corresponding family of curvature forms.Let A(s) = dω(s)/ds.A formula of Chern and Simons states:-Γ/(ω(0)) = k(k -\)d jV(s)ds + k J'/WW, Ω(s), , Ω(s))ds 0 0V(s) = /(JO), tω(s), Ω(s) t , , Ω(s) t )dt .0 where