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

Abstract

In this paper, we prove a local index theorem for the DeRham Hodge-Laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.2000 Mathematics Subject Classification. 58J20.

A stochastic Gauss–Bonnet–Chern formula

“…Still in the case E = T M , one can use rather different probabilistic ideas (Malliavin calculus) to prove the cohomological Gauss-Bonnet; the case when ∇ is the LeviCivita connection of a metric on M was investigated by E. Hsu [12], while the case of a general metric connection on T M was recently investigated by H. Zhao [21]. comments, suggestions and questions that helped improve the quality of this paper.…”