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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.

MSC:
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
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