The geodesic flow on a Riemannian supermanifold

TL;DR

This paper extends the concept of geodesics and the geodesic flow to Riemannian supermanifolds, defining a superflow on the cotangent bundle that corresponds to geodesics and generalizes classical geometric structures.

Contribution

It introduces a natural definition of geodesics on Riemannian supermanifolds and constructs a superflow and exponential map, extending classical Riemannian geometry to the supermanifold setting.

Findings

Defined geodesics on Riemannian supermanifolds.

Constructed a supergeodesic flow on the cotangent bundle.

Generalized the exponential map and isometry linearization.

Abstract

We give a natural definition of geodesics on a Riemannian supermanifold and extend the usual geodesic flow defined on the cotangent bundle of the body of the supermanifold, associated to the induced Riemannian structure on the body, to a geodesic "superflow" on the cotangent bundle of the supermanifold. Integral curves of this flow turn out to be in natural bijection with geodesics on the Riemannian supermanifold. We also construct the corresponding exponential map and generalize the well-known faithful linearization of isometries to Riemannian supermanifolds.