Grassmann Electrodynamics and General Relativity

Details Grassmann Electrodynamics and General Relativity Grassmann Electrodynamics and General Relativity

2003-09-23

arxiv.org/abs/math/0309376v1

Journal of Geometry and Physics, Vol. 51, No. 2 (2004), s. 196-210

Mathematics

Differential Geometry

16 pages. submitted to Journal of Geometry and Physics

Scientific paper

10.1016/j.geomphys.2003.10.006

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.

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