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Champs de vecteurs, flots et géodésiques sur les supervariétés

Abstract : We give a natural definition of geodesics on a Riemannian supermanifold $(\ca, g)$ and extend the usual geodesic flow on $T^*M$ associated to the underlying Riemannian manifold $(M,g)$ to a geodesic "superflow" on $T^*\ca$. Integral curves of this flow turn out to be in natural bijection with geodesics on $\ca$. We also construct the corresponding exponential map and generalize the well-known faithful linearization of isometries to Riemannian supermanifolds. We give also a new proof of the Monderde et al. result about flows of non-homogeneous supervector fields. We give a treatment which allows extensions for instance to the holomorphic category. The original proof given by Monderde et al. is only applicable to split supermanifolds, since their proofs relied on Batchelor's Theorem. Finally, we reproves a characterization of vector fields whose flows are local $\sbb$-actions of an appropriate Lie supergroups structure
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https://hal.univ-lorraine.fr/tel-01749209
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Stéphane Garnier. Champs de vecteurs, flots et géodésiques sur les supervariétés. Mathématiques générales [math.GM]. Université de Lorraine, 2012. Français. ⟨NNT : 2012LORR0040⟩. ⟨tel-01749209⟩

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