Several new actions of $p$-branes based on bulk scalar fields

Generally, $p$-branes play central roles in revealing the nonperturbative structures of the string/M-theory. In this paper, through a topological tensor current constructed in terms of a bulk scalar field, we first show some topological properties of $p$-branes, and then mainly discuss the construction of several new actions of $p$-branes. We show that the actions we construct which can be reduced to the Nambu-Goto action or some other simple actions, are all defined naturally in the bulk spacetime, preserving the full spacetime Lorentz invariance and satisfying a self-dual condition.