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A lossless reduction of geodesics on supermanifolds to non-graded differential geometry. (English) Zbl 1340.58005
By the Batchelor’s theorem, every supermanifold $$\mathcal{M}$$ is (noncanonically) isomorphic to a supermanifold $$\Pi E$$ for a vector bundle $$E \to X$$. Given a Riemannian metric $$g$$ with the Levi-Civita connection $$\nabla$$ on $$\mathcal{M}$$, the isomorphism $$\mathcal{M} \cong \Pi E$$ naturally induces well-defined objects $$g^{TE}$$ (a symmetric bilinear form) and $$\nabla^{TE}$$ (a connection compatible with $$g^{TE}$$) on $$\Pi E$$. Authors further show that there is a natural identification between geodesics for $$(\mathcal{M},\nabla)$$ (as maps $$\mathbb{R}^{1|1} \to \mathcal{M}$$) and geodesics for $$(E, \nabla^{TE})$$.

##### MSC:
 58A50 Supermanifolds and graded manifolds 53C22 Geodesics in global differential geometry 53B21 Methods of local Riemannian geometry 53C05 Connections (general theory)
##### Keywords:
supermanifolds; geodesics; Riemannian metrics; connections
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