arXiv subjects

Raphaël Alexandre

Publications and source records attributed to Raphaël Alexandre.

5 recordsLinked to original sources

arXiv2021

Closed ray nil-affine manifolds and parabolic geometries

Ray nil-affine geometries are defined on nilpotent spaces. They occur in every parabolic geometry and in those cases, the nilpotent space is an open dense subset of the corresponding flag manifold. We are interested in closed manifolds having a ray nil-affine structure. We show that under a rank one condition on the isotropy, closed manifolds are either complete or their developing map is a cover onto the complement of a nil-affine subspace. We prove that if additionally there is a parallel volume or if the automorphism group acts non properly then closed manifolds are always complete.This paper is a sequel to a previous work on ray manifolds in affine geometry.

math.DG↗

arXiv2021

Redundancy of triangle groups in spherical CR representations

Falbel, Koseleff and Rouillier computed a large number of boundary unipotent CR representations of fundamental groups of non compact three-manifolds. Those representations are not always discrete. By experimentally computing their limit set, one can determine that those with fractal limit sets are discrete. Many of those discrete representations can be related to (3,3,n) complex hyperbolic triangle groups. By exact computations, we verify the existence of those triangle representations, which have boundary unipotent holonomy. We also show that many representations are redundant: for n fixed, all the (3,3,n) representations encountered are conjugate and only one among them is uniformizable.

math.DG↗

arXiv2021

Closed ray affine manifolds

We consider closed manifolds that possess a so called rank one ray structure. That is a (flat) affine structure such that the linear part is given by the products of a diagonal transformation and a commuting rotation. We show that closed manifolds with a rank one ray structure are either complete or their developing map is a cover onto the complement of an affine subspace. This result extends the geometric picture given by Fried on closed similarity manifolds. We prove, in the line of Markus conjecture, that if the rank one ray geometry has parallel volume, then closed manifolds are necessarily complete. Finally, we show that the automorphism group of a closed manifold acts non properly when the manifold is complete.

math.DG↗

arXiv2020

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.

math.DG↗

arXiv2019

Fried's theorem for boundary geometries of rank one symmetric spaces

After introducing the different boundary geometries of rank one symmetric spaces, we state and prove Fried's theorem in the general setting of all those geometries: a closed manifold with a similarity structure is either complete or the developing map is a covering onto the Heisenberg-type space deprived of a point.

math.DG↗