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Starting from the usual bosonic membrane action, we develop the geometrysuitable for the description of $p$-brane backgrounds. Using the tools ofgeneralized geometry we derive the generalization of string open-closedrelations. Nambu-Poisson structures are used to generalize the concept ofsemiclassical noncommutativity of $D$-branes governed by Poisson tensor. Wenaturally describe the correspondence of recently proposed commutative andnoncommutative versions of an effective action for $p$-branes ending on a$p'$-brane. We calculate the power series expansion of the action in backgroundindependent gauge. Leading terms in the double scaling limit are given by ageneralization of a (semi-classical) matrix model.

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