A Lossless Reduction of Geodesics on Supermanifolds to Non-graded Differential Geometry

LetM = (M,OM) be a smooth supermanifold with connection ∇ and Batchelor model OM ∼= ΓΛE∗ . From (M,∇) we construct a connection on the total space of the vector bundle E →M . This reduction of∇ is well-defined independently of the isomorphism OM ∼= ΓΛE∗ . It erases information, but however it turns out that the natural identification of supercurves inM (as maps from R1|1 toM) with curves in E restricts to a 1 to 1 correspondence on geodesics. This bijection is induced by a natural identification of initial conditions for geodesics onM, resp. E. Furthermore a Riemannian metric on M reduces to a symmetric bilinear form on the manifold E. Provided that the connection on M is compatible with the metric, resp. torsion free, the reduced connection on E inherits these properties. For an odd metric, the reduction of a Levi-Civita connection on M turns out to be a Levi-Civita connection on E.