Topological structure of Gauss-Bonnet-Chern theorem and -branes

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

Topological structure of Gauss--Bonnet--Chern theorem and p-branes

By making use of the $phi$-mapping topological current theory, thispaper shows that the Gauss--Bonnet--Chern density (theEuler--Poincar'{e} characteristic $chi(M)$ density) can beexpressed in terms of a smooth vector field ${bm phi}$ and takethe form of $delta(bm phi)$, which means that only the zeros of$bm phi$ contribute to $chi(M)$. This is the elementary fact ofthe Hopf theorem. Furthermore, it presents that a new topologicaltensor current of $tilde {p}$-branes can be derived from theGauss--Bonnet--Chern density. Using this topological current, itobtains the generalized Nambu action for multi $tilde p$-branes.