Geodesics on a supermanifold and projective equivalence of super connections

TL;DR

This paper extends the classical theory of geodesics and projective equivalence to supermanifolds, defining supergeodesics and establishing conditions for projective equivalence of superconnections.

Contribution

It introduces a definition of supergeodesics via tangent bundle projections and proves a super version of Weyl's characterization of projective equivalence.

Findings

Supergeodesics are projections of integral curves of a vector field on the tangent bundle.

Supergeodesics coincide with free particle trajectories when the connection is metric.

Two torsion-free superconnections are projectively equivalent if their difference tensor is expressed by a super 1-form.

Abstract

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.