Heat trace asymptotics and the Gauss–Bonnet theorem for general connections

Abstract

Abstract. We examine the local super trace asymptotics for the de Rham complex defined by an arbitrary super connection on the exterior algebra. We show, in contrast to the situation in which the connection in question is the Levi-Civita connection, that these invariants are generically non-zero in positive degree and that the critical term is not the Pfaffian.

Heat trace asymptotics and boundedness in the second order Sobolev space of isospectral potentials for the Dirichlet Laplacian

“…As a matter of fact Dryden, Gordon, Greenwald and Webb recently calculated the asymptotic expansion of the heat kernel for orbifolds in [12]. In the same spirit, the heat trace asymptotics for general connections has been expressed by Beneventano, Gilkey, Kirsten and Santangelo in [2].…”

A note on the Gauss–Bonnet–Chern theorem for general connection

“…The fundamental solution's asymptotic expansion in (x, x) is determined by local condition around x (see [4] chapter 2). So we may assume M is spin manifold, whose spinor bundle is denoted by S, dual bundle is denoted by S * .…”

“…In the above theorem, the DeRham operator and Hodge-Laplacian are defined by Levi-Civita connection. Recently, Beneventano-Gilkey-Kirsten-Santangelo [4] have studied the Gauss-Bonnet theorem for general connection and corresponding heat trace's asymptotic expansion. Bell [5] has given Gauss-Bonnet theorem for vector bundle whose rank is equal to the dimension of underlying manifold.…”