Odd symplectic geometry types on supermanifolds

Generalization of symplectic geometry on manifolds in a supersymmetric case is examined in the present work. In the even case, this leads either to even symplectic geometry, that is, the geometry on supermanifolds with the nondegenerate Poisson bracket, or to the geometry on the Fedosov even supermanifolds. In the odd case, two different scalar symplectic structures exist (namely, the odd closed differential 2-form and antibracket), which can be used to construct various symplectic geometry types on supermanifolds. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 52–57, February, 2008.