  We define a new one form H^A based on the second fundamental tensor H^abA,
the Gauss-Bonnet-Chern form can be novelly expressed with this one-form. Using
the phi-mapping theory we find that the Gauss-Bonnet-Chern density can be
expressed in terms of the delta-function and the relationship between the
Gauss-Bonnet-Chern theorem and Hopf-Poincare theorem is given
straightforwardly. The topological current of the Gauss-Bonnet-Chern theorem
and its topological structure are discussed in details. At last, the Morse
theory formula of the Euler characteristic is generalized.
