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Geodesic convexity and closed nilpotent similarity manifolds


Some nilpotent Lie groups possess a transformation group analogous to the similarity group acting on the Euclidean space. We call such a pair a nilpotent similarity structure. It is notably the case for all Carnot groups and their dilatations. We generalize a theorem of Fried: closed manifolds with a nilpotent similarity structure are either complete or radiant and, in the latter case, complete for the structure of the space deprived of a point. The proof relies on a generalization of convexity arguments in a setting where, in the coordinates given by the Lie algebra, we study geodesic segments instead of linear segments. We show classic consequences for closed manifolds with a geometry modeled on the boundary of a rank one symmetric space.
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hal-02500719 , version 1 (06-03-2020)



Raphaël V Alexandre. Geodesic convexity and closed nilpotent similarity manifolds. 2020. ⟨hal-02500719⟩
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