arXiv research

Study on Riemannian Poisson warped product spaces and their properties.

We survey results on compact Clifford-Klein forms of homogeneous spaces, with a focus on recent contributions and organized around approaches via topology, geometry and dynamics. In addition, we survey results on moduli spaces of compact forms.

A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2,n), such that G/H has a compact Clifford-Klein…

Sharp bounds on quasimode norms on compact space forms.

Classifies compact Clifford-Klein forms for specific Lie algebras.

Paper finds obstructions to compact Clifford-Klein forms for tangential symmetric spaces.

Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.

We calculate the rational equivariant cohomology of the spaces of non-contractible loops in compact space forms and show how to apply these calculations for proving the existence of closed geodesics.

Classifies 4D spaces with compact Clifford-Klein forms.

This article continues a line of research aimed at solving an important problem of T. Kobayashi of the existence of compact Clifford-Klein forms of reductive homogeneous spaces. We contribute to this topic by showing that almost all symmetric spaces and 3-symmetric spaces do not admit solvable compact CliffordfKlein fo…

The Hodge decomposition is well-known for compact manifolds. The result has been extended by Kodaira to include non-compact manifolds and $L^2$ forms. We further extend the Hodge decomposition to the Sobolev space $H^1$ for general $k$-forms on non-compact manifolds of nonpositive constant sectional curvature. As a res…

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjectu…

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

Solves classification of compact Clifford-Klein forms for specific Lie groups.

We provide a necessary condition for the existence of a compact Clifford-Klein form of a given homogeneous space of reductive type. The key to the proof is to combine a result of Kobayashi-Ono with an elementary fact that certain two different Clifford-Klein forms have the same cohomology ring. We give some examples, S…

The study examines harmonic forms on almost Hermitian manifolds and complex surfaces.

Researchers generalize space forms in Riemannian geometry using specific vector fields.

Let $M=S^{2n+1}/ Γ$, $Γ$ is a finite group which acts freely and isometrically on the $(2n+1)$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we first investigate Katok's famous example about irreversible Finsler metrics on the spheres to study the topological structure of the contrac…

Paper proves new rigidity results for biconservative hypersurfaces.

This article studies the volume of compact quotients of reductive homogeneous spaces. Let $G/H$ be a reductive homogeneous space and $Γ$ a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We prove that the volume of $Γ\backslash G/H$ is the integral, over a certain homology class of $Γ…

In this article we prove that under certain assumptions, a reductive homogeneous space G/H does not admit a solvable compact Clifford-Klein form. This generalizes the well known non-existence theorem of Benoist for nilpotent Clifford-Klein forms. This generalization works for a particular class of homogeneous spaces de…

New findings on Kähler-Einstein currents dominating Kähler forms on compact spaces.

Harmonic 3-forms defined on compact homogeneous spaces are studied and conditions for their harmonicity are provided.

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space $G/H$ if $H$ is non-compact. The first half of the article elucidates general machinery …

Study finds rigidity of biconservative hypersurfaces in space forms without curvature assumptions.

This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, technique…

Derives integral formula for differential forms on compact spaces with applications.

Let $M=S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n\geq2$, $Γ$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractibl…

We use a computer-aided approach to prove that there are no standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups. This yields further support for Kobayashi's conjecture about possible compact Clifford-Klein forms. On one hand, our approach is based on the algorithms developed in this wo…

Lyapunov 1-forms on orbifolds help understand flows on compact spaces.

Study submanifolds in curved spaces with specific curvature bounds.

Study primitive decompositions for harmonic forms on almost Kähler manifolds.

The paper proves conditions for the existence of multiple non-contractible closed geodesics on Finsler compact space forms.

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 study finds non-contractible geodesics on compact Finsler space forms without intersections.

The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.

S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its pert…

In this note we show that any real exact G-invariant (1,1)-form is the Ricci form of a Kaehler metric on the complexification of an irreducible compact symmetric space G/K.

In an earlier paper we showed that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. In that paper, we also introduced a new class of Lagrangian-type 4-dimensional su…

Study immersions of Sasakian manifolds into Sasakian space forms.

Study on surfaces pinched by curvature in space forms converging under specific conditions.

New characterizations for manifolds with boundary rigidity results.

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew--symmetric. We show that a compact simply connected symmetric space carries a non--parallel Killing $p$--form ($p\ge2$) if and only if it isometric to a Riemannian product $S^k\times N$, where $S^k$ is a round sphere…

Let $(X,g)$ be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of $X$, as well as the associated spac…

The paper introduces new cohomologies and studies harmonic forms on almost complex manifolds.

Study quadratic one-forms on logarithmic Higgs bundles on pointed curves.

The paper studies harmonic 1-forms on specific metric measure spaces.

Study on eigenvalues of Laplace operator on 1-forms for symmetric spaces.