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

TL;DR

This paper extends the classical Gauss-Bonnet-Chern theorem to a broader context by proving a local index theorem for the DeRham Hodge-Laplacian associated with general metric-compatible connections, not limited to Levi-Civita.

Contribution

It introduces a local index theorem for the DeRham Hodge-Laplacian with arbitrary metric-compatible connections, generalizing the classical theorem.

Findings

Proves a local index theorem for the DeRham Hodge-Laplacian with general connections.

Shows the classical Gauss-Bonnet-Chern theorem as a special case when using Levi-Civita 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.