Decomposition Theory of Spin Connection and Topological Structure of Gauss-Bonnet-Chern Theorem on Manifold With Boundary

The index theorem of Euler-Poincar\'e characteristic of manifold with boundary is given by making use of the general decomposition theory of spin connection. We shows the sum of the total index of a vector field $\phi $ and half the total of the projective vector field of $\phi $ on the boundary equals the Euler-Poincar\'e characteristic of the manifold. Detailed discussion on the topological structure of the Gauss-Bonnet-Chern theorem on manifold with boundary is given. The Hopf indices and Brouwer degrees label the local structure of the Euler density.