arXiv research

Projective structures are mostly rigid at the boundary but some are not.

We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?

We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spac…

We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.

Study shows Einstein structures on 4-manifolds are rigid.

Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.

Projective rigidity of circle packings on complex surfaces proved.

The study proves rigidity for mixed Hodge structures and applies to curve families.

The paper shows inequality and rigidity for manifolds with integral Ricci curvature.

Rigorous model for 2-gerbes simplifies calculations in physics.

New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.

New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.

Unified framework for rigidity results on $(κ, μ)$-manifolds.

Paper proves rigidity of discrete conformal structures on polyhedral surfaces.

Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.

This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains.

Counterexample shows ADC contact structures can't have isomorphic cohomologies.

We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

The classical Liouville Theorem on conformal transformations determines local conformal transformations on the Euclidean space of dimension $\geq 3$. Its natural adaptation to the general framework of Riemannian structures is the 2-rigidity of conformal transformations, that is such a transformation is fully determined…

New method for flexible tubes and structures, enabling rigid-foldability.

To a hyperbolic manifold one can associate a canonical projective structure and ask whether it can be deformed or not. In a cusped manifold, one can ask about the existence of deformations that are trivial on the boundary. We prove that if the canonical projective structure of a cusped manifold is infinitesimally proje…

We study the moduli space of quaternionic Kaehler structures on a compact manifold of dimension 4n (n>2) from a point of view of Riemannian geometry, not twistor theory. Then we obtain a rigidity theorem for quaternionic Kaehler structures of nonzero scalar curvature by observing the moduli space.

Study second-order obstruction to nearly $G_2$ structure deformations.

Following a survey of the abstract boundary definition of Scott and Szekeres, a rigidity result is proved for the smooth case, showing that the topological structure of the regular part of this boundary in invariantly defined.

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbo…

Survey on harmonic maps in non-smooth spaces, focusing on rigidity.

Study on mappings in Carnot groups, proving rigidity results.

Study constructs $H$-space on metric spaces, providing rigidity criteria.

A new method for non-rigid point set registration reduces computational complexity.

Study proves structure results for homogeneous spaces supporting specific equations.

For 3-dimensional hyperbolic cone structures with cone angles $θ$, local rigidity is known for $0 \leq θ\leq 2π$, but global rigidity is known only for $0 \leq θ\leq π$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $π$ do not degenerate in defo…

In the present paper we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions …

New 2D complex hyperbolic structures found on sphere orbibundles.

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, t…

Develops computational methods for simulating rigid body dynamics on SO(3).

Rigidity theorem for flag manifolds in various dimensions.

The paper studies Einstein-type manifolds with structural conditions.

In this paper, we prove a rigidity theorem of asymptotically hyperbolic manifolds only under the assumptions on curvature. Its proof is based on analyzing asymptotic structures of such manifolds at infinity and a volume comparison theorem.

Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric st…

The paper describes complex structures on Oeljeklaus-Toma manifolds.

This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give sufficient conditions for local homogeneity in a broad class of such structures, namely Cartan geometries, extending a classical result of Singer about locally homogeneous Riemanni…

The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.

Study of infinitesimal rigidity in hyperbolic manifolds.

Paper proves rigidity of de-Sitter tori with conical singularities.

Study uses Zilber-Pink conjecture and dynamical methods to solve rigidity problems.

For an $n$-dimensional real hyperbolic manifold $M$, we calculate the Zariski tangent space of a character variety $χ(π_1(M),SL(n+1,\mathbb R)), n>2$ at Fuchisan loci to show that the tangent space consists of cubic forms. Furthermore we prove the Weil's local rigidity theorem for uniforml hyperbolic lattices using rea…

In this article, we generalize Eberlein's Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible locally symmetric space of noncompact type does not admit a nonpositively curved (in t…

Homotopy operators help describe structures in equivariant deformation problems.