The geodesic flow on a Riemannian supermanifold

We give a natural definition of geodesics on a Riemannian supermanifold (M,g) and extend the usual geodesic flow on T∗M associated to the underlying Riemannian manifold (M,g˜) to a geodesic “superflow” on T∗M. Integral curves of this flow turn out to be in natural bijection with geodesics on M. We also construct the corresponding exponential map and generalize the well-known faithful linearization of isometries to Riemannian supermanifolds.