A note on the Gauss-Bonnet-Chern theorem for general connection
  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. 
Haoyan Zhao
math.DG math-ph math.MP