arXiv research

The paper proves a theorem about earthquake extensions of vector fields on circles.

Study surfaces in Half-Pipe space and vector fields on hyperbolic plane.

Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.

We highlight the relation between the projective geometries of $n$-dimensional Euclidean, spherical and hyperbolic spaces through the projective models of these spaces in the $n+1$-dimensional Minkowski space, using a cross ratio notion which is proper to each of the three geometries.

An axiomatic characterization of buildings of type $\CC_3$ due to Tits is used to prove that any cohomogeneity two polar action of type $\CC_3$ on a positively curved simply connected manifold is equivariantly diffeomorphic to a polar action on a rank one symmetric space. This includes two actions on the Cayley plane w…

We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph $Γ$ is realized as the $1$-skeleton of a polyhedron inscribed in the hyperboloid or cyl…

Study character varieties of a Coxeter group in hyperbolic and Anti-de Sitter spaces.

Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a functional introduced by Bonsante, Mondello \& Schlenker, we show that there exists a …

Study on BAS manifolds with parallel torsion and curvature.

Given an orientable ideally triangulated $3$--manifold $M$, we define a system of real valued equations and inequalities whose solutions can be used to construct projective structures on $M$. These equations represent a unifying framework for the classical Thurston gluing equations in hyperbolic geometry and their more…

We provide the first examples of geometric transition from hyperbolic to anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional…

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path i…

The paper provides examples of geometric transitions in low dimensions.

Paper constructs geometric transitions between hyperbolic and Anti-de Sitter geometries.

We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra t…

We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces and tori. We give proofs of uniqueness anc characterize non-uniqueness in different settings. Reconstruction …

We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24(g-1),…

Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization…

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

Part I. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method…

Since the end of the 19th century, and after the works of F. Klein and H. Poincaré, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen as a "limit" of both geometries. Then all the geometries that can be obtained in t…

The Weyl problem is extended to hyperbolic and anti-de Sitter spaces, connecting geometry, analysis, and group theory.

Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.