A lossless reduction of geodesics on supermanifolds to non-graded differential geometry

Abstract

Let $\mathcal{M}=(M,{\mathcal{O}}_{\mathcal{M}})$ be a smooth supermanifold with connection $\nabla$ and Batchelor model ${\mathcal{O}}_{\mathcal{M}}\cong {\Gamma }_{\Lambda E^{\ast }}$. From $(\mathcal{M},\nabla )$ we construct a connection on the total space of the vector bundle $E\to M$. This reduction of $\nabla$ is well-defined independently of the isomorphism ${\mathcal{O}}_{\mathcal{M}}\cong {\Gamma }_{\Lambda E^{\ast }}$. It erases information, but however it turns out that the natural identification of supercurves in $\mathcal{M}$ (as maps from ${\mathbb{R}}^{1|1}$ to $\mathcal{M}$) 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 on $\mathcal{M}$, resp. $E$. Furthermore a Riemannian metric on $\mathcal{M}$ reduces to a symmetric bilinear form on the manifold $E$. Provided that the connection on $\mathcal{M}$ 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 $\mathcal{M}$ turns out to be a Levi-Civita connection on $E$.