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Geodesic convexity and closed nilpotent similarity manifolds

Abstract : 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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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-02500719
Contributor : Raphaël Alexandre <>
Submitted on : Friday, March 6, 2020 - 11:57:43 AM
Last modification on : Wednesday, December 9, 2020 - 3:06:10 PM
Long-term archiving on: : Sunday, June 7, 2020 - 1:26:29 PM

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  • HAL Id : hal-02500719, version 1
  • ARXIV : 2003.03169

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

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