Grassmann electrodynamics and general relativity

Abstract

The aim of this paper is to present a short introduction to supergeometry on pure odd supermanifolds. (Pseudo)differential forms, Cartan calculus (DeRham differential, Lie derivative, interior product), metric, “inner” product, Killing’s vector fields, Hodge star operator, integral forms, co-differential and connection on odd Riemannian supermanifolds are introduced. The electrodynamics and Einstein relativity with anti-commuting variables only are formulated modifying the geometry beyond classical (even, bosonic) theories appropriately. Extension of these ideas to general supermanifolds is straightforward.