A Bismut type theorem for subelliptic heat semigroups

Abstract

Given a general second order subelliptic differential operator L defined on a vector bundle E over a compact manifold, we study the existence of lim t → 0 σ ( p t ( x , x ) ) , where p t is the heat kernel of e t L and σ is a linear map on End ( E x ) . Our result contains as a special case the local Atiyah–Singer index theorem for Dirac operators on Clifford bundles. Our approach is based on an extension to fiber bundles of the link pointed out by Rotschild and Stein between Nilpotent Lie groups and subelliptic heat kernel asymptotics on the diagonal. To cite this article: F. Baudoin, C. R. Acad. Sci. Paris, Ser. I 344 (2007).