arXiv research

New findings show fundamental group is not audible in spherical space forms.

New curvature positivity helps classify spherical spaces and complex projective spaces.

Researchers find highest volumes for isospectral spherical orbifolds and space forms.

The study characterizes compact submanifolds with pinched Ricci curvature in Euclidean and spherical space forms.

Free actions of finite groups on spheres give rise to topological spherical space forms. The existence and classification problems for space forms have a long history in the geometry and topology of manifolds. In this article, we present a survey of some of the main results and a guide to the literature.

The paper extends geometric inequalities for nearly spherical sets in various space forms.

New flow for capillary surfaces converges to spherical caps.

The paper proves stability of inequalities for nearly spherical sets in various spaces.

We prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two. This confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.

Dunkl connections on complex plane don't preserve metrics.

New spherical T-duality for higher degree forms in fiber bundles.

Sharp curvature condition implies spherical space form structure.

We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.

Let $M$ be a topological spherical space form, i.e. a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

The study explores how to infer the geometry of space forms from similarity comparisons.

Study curvature operator on Riemannian manifolds, proving new classification results.

We prove that closed manifolds admitting a generic metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC spaces where sets of isotropic points are arbitrary, under suitable positivity assumption and for torsion-free fundamental groups they are still diff…

P. Baird and the second author studied harmonic morphisms from a three-dimensional simply-connected space form to a surface and obtained a complete local and global classification of them. In this paper, we obtain a description of all harmonic morphisms from any three-dimensional Euclidean and spherical space form to a…

The paper proves conditions for a manifold to be homeomorphic to a spherical space form.

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space $E^n = G/K$ where $G$ is the semidirect product $R^n \cdot K$ of the translation group with a closed subgroup $K$ of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K…

We show that every spherical 2-Dupin submanifold that is not a hypersurface is conformally congruent to the standard embedding of the real, complex, quaternionic or octonionic projective plane. We also classify 2-CPC, 2-umbilical and weakly 2-umbilical submanifolds in space forms.

New inequalities for convex hypersurfaces in various spaces.

It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional at compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.

The analytic torsion is computed on fixed-point free and non fixed-point free factors (tessellations) of the three--sphere. We repeat the standard computation on spherical space forms (Clifford-Klein spaces) by an improved technique. The transformation to a simpler form of the spectral expression of the torsion on sphe…

Study the Hessian geometry of an ideal gas in a centrifuge.

The paper verifies deep neural networks' ability to approximate functions on spheres.

Study of pulleys and gears in spherical and hyperbolic geometries.

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

Survey on 4-manifolds with specific curvature properties.

A new spherical Sliced-Wasserstein distance for data on spheres.

Study spherical doubly warped spacetimes for stellar collapse and cosmology.

Proposes a new latent variable model for hyperspherical latent spaces.

The study connects curvature operators' positivity to manifold topology.

We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the $L^2$-norm of their scalar curvature and…

In this note, we first give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group. Moreover, we …

The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.

The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.

We construct closed symplectic manifolds for which spherical classes generate arbitrarily large subspaces in 2-homology, such that the first Chern class and cohomology class of the symplectic form both vanish on all spherical classes. We construct both Kaehler and non-Kaehler examples, and show independence of the cond…

In this paper we describe recent results on explicit construction of lens spaces that are not strongly isospectral, yet they are isospectral on $p$-forms for every $p$. Such examples cannot be obtained by the Sunada method. We also discuss related results, emphasizing on significant classical work of Ikeda on isospectr…

We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms.

Abstract: Generalizes SGMs to infinite-dimensional Hilbertian setting.

Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.

It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed…

We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the c…

The study proves rigidity and non-rigidity of spherical caps in mean curvature.

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere…

New classification for certain compact manifolds with positive isotropic curvature.