  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.
