arXiv research

Researchers found a maximal antipodal set of three elements in a 7x7 sphere space.

Classification of 3-symmetric spaces with Ricci solitons.

We describe complex twistor spaces over inner 3-symmetric spaces $G/H$, such that $H$ acts transitively on the fibre. Like in the symmetric case, these are flag manifolds $G/K$ where $K$ is the centralizer of a torus in $G$. Moreover, they carry an almost complex structure defined using the horizontal distribution of t…

The structure of nearly Kähler manifolds was studied by Gray in several papers. More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a strict and complete nearly Kähler manifold is locally a Riemannian product of homogeneous nearly Kähler spaces, twistor spaces ov…

R. Lutz introduced the notion of $\varGamma$-symmetric space as a generalization of the classical notion of symmetric space in 1981, where $\varGamma$ is a finite abelian group. In the present article, as $\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$, we give the automorphisms $\tildeσ_3, \tildeτ_3$ of order $3$ o…

Bielavsky introduced and investigated the class of symmetric symplectic spaces, that is, symmetric spaces endowed with a symplectic form invariant with respect to symmetries. Since the theory of symmetric spaces has generalizations, we ask a question about their possible symplectic versions. We do construct such genera…

We study 6-dimensional nearly Kahler manifolds admitting a Killing vector field of unit length. In the compact case it is shown that up to a finite cover there is only one geometry possible, that of the 3--symmetric space $S^3 \times S^3$.

We show that any non-Kahler, almost Kahler 4-manifold for which both the Ricci and the Weyl curvatures have the same algebraic symmetries as they have for a Kahler metric is locally isometric to the (only) proper 3-symmetric 4-dimensional space described by O. Kowalski.

In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped with a homogeneous spin structure can be viewed as the simplest manifolds after Riemannian symmetric spin spaces. In this paper, we give some characterizations and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds and …

We classify six-dimensional homogeneous nearly Kähler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly Kähler manifold is a Riemannian 3-symmetric space equipped with its canonical almost Hermitian structure. The only four examples in dimension 6 are $S^3 \times S^3$, the com…

We prove Gray & Wolf's conjecture that a Riemannian homogeneous manifold admitting a strict nearly Kahler structure is 3-symmetric. We actually classify them in dimension 6 and use previous results of Swann, Cleyton and Nagy to prove the conjecture in higher dimensions.

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…

We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds F_n=SU(n+2)/S(U(n)\times U(1)\times U(1)). For all n>1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles …

New exact spherically symmetric vacuum solutions found in Finsler gravity.

The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle ${\cal P}$. The structure group G=SO(3) is the vorticity group, and the bundle ${\cal P}=GL_+(3, R})$ is the connected component of the general linear group. The base manifold is the space of positive-defi…

We construct a black hole initial data for the Einstein equations with prescribed scalar curvature, or more precisely a piece of initial data contained inside the black hole. The constraints translate into a parabolic equation, with radius as "time" variable, on a metric component u that undergoes blow up. The metric i…

Study confirms conjectures on Ricci limit spaces and their topological properties.

The space of tensors of metric curvature type on a Euclidean vector space carries a two-parameter family of orthogonally invariant commutative nonassociative multiplications invariant with respect to the symmetric bilinear form determined by the metric. For a particular choice of parameters these algebras recover the p…

New 1-parameter family of ovals identified in 4d Ricci flow classification.

Proposes a normalization technique for manifold valued data.

New findings on tensor decomposition complexity, showing polynomial functions can estimate the largest component under certain conditions.

Study of tangent spaces in diffeological spaces under Lie group actions.

(1,1) non-L-space knots are foliar in 3D space.

Universal spaces for finite topological spaces simplify shape descriptions.

The paper extends Stone duality to topological convexity spaces.

No Einstein hypersurfaces found in Damek-Ricci spaces.

The distance function $\varrho(p,q)$ (or $d(p,q)$) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over $\mathbb R^n$, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…

New kernels defined for various spaces, including measures.

Metric spaces uniquely split into Hilbert and non-line-split parts.

In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …

Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.

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

Study of complete space-like self-expanders in Minkovski space.

The abstract discusses the linear and smooth structures of mapping spaces.

Study on convergence of transformed metric spaces as dimensions grow.

Characterizes when almost smooth spaces become RCD spaces.

Stability of Wasserstein spaces under various convergence types.

New infinite families of flat spaces found from symmetric spaces.

New quasi space forms solve Thurston's geometrical space form problem.

Classifies compact spaces by shape, finite spaces by weak homotopy.

Estimates eigenvalues of Jacobi operator for harmonic spaces.

Introduces new types of homogeneous spaces and their properties.

This work describes compactifications of metric spaces and vector spaces using asymmetric norms.

In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

Cantor Riemannium is a new type of space from holomorphic germs.

We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplect…