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

TL;DR

This paper presents a method to reduce the study of geodesics on supermanifolds to classical differential geometry on vector bundles, establishing a one-to-one correspondence of supercurves and geodesics.

Contribution

It introduces a natural reduction process for connections on supermanifolds that preserves geodesic structures and metric properties, bridging supergeometry with classical differential geometry.

Findings

Supercurves in supermanifolds correspond bijectively to curves in associated vector bundles.

The reduction preserves metric compatibility and torsion-free properties.

Levi-Civita connections reduce to Levi-Civita connections under the process.

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 $E$.