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 $E$.