General Decomposition of Spin Connection and Topological Structure in Gauss-Bonnet-Chern Density

By means of devices of geometric algebra the general decomposition of spin connection on the sphere bundle of a compact n-dimensional Riemannian manifold has been studied in detail. Using this decomposition theory it is shown that the Gauss-Bonnet-Cherniensity of the Euler-PoincarB characterictic can be expressed in terms of a smooth vector field φ and take the form of the 6 function δ(φ). The topological structure in Gauss-Bonnet-Chern theorem is reviewed.