Extended generalized geometry and a DBI-type effective action for branes ending on branes

Starting from the usual bosonic membrane action, we develop the geometry suitable for the description of $p$-brane backgrounds. Using the tools of generalized geometry we derive the generalization of string open-closed relations. Nambu-Poisson structures are used to generalize the concept of semiclassical noncommutativity of $D$-branes governed by Poisson tensor. We naturally describe the correspondence of recently proposed commutative and noncommutative versions of an effective action for $p$-branes ending on a $p'$-brane. We calculate the power series expansion of the action in background independent gauge. Leading terms in the double scaling limit are given by a generalization of a (semi-classical) matrix model.