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The geodesic flow on a Riemannian supermanifold. (English) Zbl 1242.53046
Based on B. Kostant [Differ. geom. Meth. math. Phys., Proc. Symp. Bonn 1975, Lect. Notes Math. 570, 177–306 (1977; Zbl 0358.53024)] and D. A. Leites [Russ. Math. Surv. 35, No. 1, 1–64 (1980; Zbl 0462.58002)], the present paper shows the existence and unicity of a supergeodesic (defined in a way different from O. Goertsches [Math. Z. 260, No. 3, 557–593 (2008; Zbl 1154.58001)] for a given initial condition. Then, a natural “energy function” H on the co-tangent bundle \(T*\mathcal M\) of a supermanifold \(\mathcal M\) and an associated Hamiltonian vector field \(X_H\) is obtained. It is proved here that integral curves of \(X_H\) on \(T*\mathcal M\) are in natural bijection with geodesics on \(\mathcal M\). The corresponding exponential map is constructed. As an example of an application it is shown that linearization in a fixed point of an isometry of a Riemannian supermanifold is faithful.

53C22 Geodesics in global differential geometry
58A50 Supermanifolds and graded manifolds
53D25 Geodesic flows in symplectic geometry and contact geometry
58C50 Analysis on supermanifolds or graded manifolds
Full Text: DOI arXiv
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