Geodesics on a supermanifold and projective equivalence of super connections

Details Geodesics on a supermanifold and projective equivalence of super connections Geodesics on a supermanifold and projective equivalence of super connections

2012-02-06

arxiv.org/abs/1202.1077v2

Mathematics

Differential Geometry

20 pages

Scientific paper

We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field associated with the connection. Our (super) geodesics possess the same properties as the in the classical case: there exists a unique (super) geodesic satisfying a given initial condition and when the connection is metric, our supergeodesics coincide with the trajectories of a free particle with unit mass. Moreover, using our definition, we are able to establish Weyl's characterization of projective equivalence in the super context: two torsion-free (super) connections define the same geodesics (up to reparametrizations) if and only if their difference tensor can be expressed by means of a (smooth, even, super) 1-form.

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