Geodesics on a supermanifold and projective equivalence of super connections

Thomas Leuther; Fabian Radoux; Gijs Tuynman

doi:10.1016/j.geomphys.2013.01.005

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arXiv · 1202.1077

Geodesics on a supermanifold and projective equivalence of super connections

Thomas Leuther·Fabian Radoux·Gijs Tuynman

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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.

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BibTeX RIS

Thomas Leuther, Fabian Radoux, Gijs Tuynman. 2012-02-06. Geodesics on a supermanifold and projective equivalence of super connections. https://doi.org/10.1016/j.geomphys.2013.01.005

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