Topological structure of Gauss--Bonnet--Chern theorem and \tildep-branes

By making use of the \phi-mapping topological current theory, this paper shows that the Gauss--Bonnet--Chern density (the Euler--Poincaré characteristic \chi(M) density) can be expressed in terms of a smooth vector field \phi and take the form of \delta(\phi), which means that only the zeros of \phi contribute to \chi(M). This is the elementary fact of the Hopf theorem. Furthermore, it presents that a new topological tensor current of \tilde p-branes can be derived from the Gauss--Bonnet--Chern density. Using this topological current, it obtains the generalized Nambu action for multi \tilde p-branes.