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.