Superstrings and supermembranes in the doubly supersymmetric geometrical approach

Abstract We perform a generalization of the geometrical approach to describing extended objects for studying the doubly supersymmetric twistor-like formulation of super p -branes. Some basic features of the embedding world supersurface into target superspace specified by a geometrodynamical condition are considered. It is shown that the main attributes of the geometrical approach, such as the second fundamental form and extrinsic torsion of the embedded surface, and the Codazzi, Gauss and Ricci equations, have their doubly supersymmetric counterparts. At the same time the embedding of the supersurface into target superspace has its particular features. For instance, the embedding may cause more rigid restrictions on the geometrical properties of the supersurface. This is demonstrated with the examples of an N = 1 twistor-like supermembrane in D = 11 and type II superstrings in D = 10, where the geometrodynamical condition causes the embedded supersurface to be minimal and puts the theories on the mass shell.