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

Let $\mathcal{M}=(M,{\mathcal{O}}_{\mathcal{M}})$ be a smooth supermanifold with connection $\nabla $ and Batchelor model ${\mathcal{O}}_{\mathcal{M}}\cong {\Gamma}_{\Lambda {E}^{*}}$. 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 ${\mathcal{O}}_{\mathcal{M}}\cong {\Gamma}_{\Lambda {E}^{*}}$. 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...