  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, "inner" 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. All this "odd business" (in both meanings of the word "odd")
is based on classical geometrical analogy.
