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IS-DAAD-Forskerutveksl. Norge-Tyskland

Degree of mobility of geometric structures and overdetermined systems of PDE

Awarded: NOK 56,885

Degree of mobility of a geometric structure is the dimension of the space of solutions of the corresponding Lie equation. The latter is usually overdetermined. We study overdetermined systems of PDE coming from differential geometry and mathematical physi cs (in particular coming from the theory of integrable systems, separation of variables and general relativity); the goal is to understand the possible values of the degrees of mobility. For many geometric structures this is a classical question and some answers are known. Namely, for Cartan geometries the maximal degree of mobility is the dimension of the corresponding Lie group. The results on sub-maximal degrees of mobility for affine connections, projective structures, 2-distributions in 5-dimensional spaces and Riemannian metrics are due to Fubini, Cartan, Egorov, Yano, Kobayashi, Nagano, Wakakuwa and Vranceanu. Recently Boris Kruglikov and Dennis The resolved the problem of computing the submaximal degree of mobility ("the first gap problem") for al l parabolic geometries, which includes conformal, Lagrangian, contact projective and exceptional geometries, generic distributions and systems of 2nd order ODEs. In parallel, Vladimir Matveev with his collaborators computed all possible degrees of mobilit y for pseudo-Riemannian metrics from the geodesic equivalence viewpoint. This corresponds to metric projective and metric c-projective structures. In this approach they greatly generalized the previous works of Sinyukov and Mikes and obtained many fascina ting global results. The first comparisons of methods lead to a joint paper on the projective and affine metric structures arXiv:1304.4426. We expect to obtain many more results as a synergy effect from combining the different approaches of two working gr oups. We have a successful story of cooperation (5 joint papers), partially thanks to DAAD grant 2011-2012. We would like to collaborate with DAAD support for the groups of 3 (professor + 2 students) from each side.

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IS-DAAD-Forskerutveksl. Norge-Tyskland