The geodesic flow on a Riemannian supermanifold

Stéphane Garnier (1) , Tilmann Wurzbacher (2)

1 LMAM - Laboratoire de Mathématiques et Applications de Metz
2 Fakultät für Mathematik [Bochum]

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 Rieman-nian 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.

Mots clés: Geodesic flow; Supermanifolds

Domaines: Géométrie différentielle [math.DG]

Stéphane Garnier, Tilmann Wurzbacher. The geodesic flow on a Riemannian supermanifold. Journal of Geometry and Physics, 2012, 62 (6), pp.1489-1508. ⟨10.1016/j.geomphys.2012.02.002⟩. ⟨hal-01279981⟩