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Pseudo-Riemannian Geometry: holonomy groups, homogeneous spaces, and foliations (PSEUDORIEMANN.BERLIN)
Start date: 01 Sep 2007, End date: 31 Aug 2009 PROJECT  FINISHED 

This project focuses on global pseudo-Riemannian geometry, i.e. geometry of manifolds with a non-degenerate, but indefinite, metric. The aim is to investigate, and if possible classify, global "special" pseudo-Riemannian geometries, in particular: (1) pseudo-Riemannian manifolds with a special holonomy group, (2) homogeneous (in particular symmetric) pseudo-Riemannian spaces, (3) foliated manifolds with a homogeneous transverse pseudo-Riemannian geometry, or with totally geodesic leaves. The main first task is (1), with focusing at first on the Lorentzian case. The point is the case where the holonomy group is not semi-simple. The three points are linked. For example, understanding (1) is helpful for (2), (2) may provide examples for (1), a work on (3) (totally geodesic leaves) may help to classify (1) in low dimension... They are also related to physical problems.To achieve the goal, several techniques are to be used, some which I know, many which I have to learn. For (1), local differential calculus, Cartan-Kahler theory, foliations, techniques developed by Galaev (analytic germs). For (2), classical Lie group theory, and techniques newly developed by Kath, Olbrich, Neukirchner (twofold extensions...). For (3), (G,X)-structures... I wish to work on those subjects with Helga Baum’s team in Berlin, Humboldt Universitat. The idea is that we have close interests, but with different, complementary approaches and competencies.That is why: - this transnational cooperation is likely to produce good results (all the more, in a subject that requires, simultaneously, several competencies), - it fits the objectives of this action, as it will complete and diversify my skills and give me an expertise in the field of and “special pseudo-Riemannian geometries”.
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