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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)
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