GENERAL DECOMPOSITION-THEORY OF SPIN CONNECTIONS TOPOLOGICAL-STRUCTURE OF GAUSS-BONNET-CHERN DENSITY AND THE MORSE-THEORY

By means of methods of the geometric algebra the general decomposition of the spin connections on the sphere bundle of a compact n-dimension al Riemannian manifold has been studied in detail. Using this decompos ition theory it is shown that the Gauss-Bonnet-Chern density of the Eu ler-Poincare characteristic can be expressed as a a-function of a smoo th vector field delta(<(phi)over right arrow>). The topological struct ure of the Gauss-Bonnet-Chern density is detailed. Furthermore the Mor se theory formula of the Euler-Poincare characteristic has been obtain ed via the topological structure.