Constructs perturbed Fefferman spaces on almost CR manifolds.
problem Characterize and construct conformal structures on almost CR manifolds.
method Introduces perturbations of Fefferman spaces using semi-basic one-forms.
result Derives conditions for conformally flat spaces on zero sets of almost Einstein scales.
An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are special cases. The governing equation is a conformally invariant overdetermined…
We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalisation of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scal…
The study finds limits on dimensions of certain scales and fields for conformal manifolds.
problem Limits on dimensions of almost Einstein scales and normal conformal Killing fields for conformal manifolds.
method Analyzes the submaximal dimensions of spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds, considering different signatures and dimensions.
result Upper bounds on dimensions of almost Einstein scales and normal conformal Killing fields are determined, with examples provided for submaximal dimensions.
In this article, we study Einstein-Weyl structures on almost cosymplectic manifolds. First we prove that an almost cosymplectic (κ,μ)-manifold is Einstein or cosymplectic if it admits a closed Einstein-Weyl structure or two Einstein-Weyl structures. Next for a three dimensional compact almost α-cosymplectic manifol…
We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non-compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with non-negative ∗-scalar curvature are cokähler…
The paper investigates (m,ρ)-quasi-Einstein structures on almost co-Kähler manifolds.
problem Investigating (m,ρ)-quasi-Einstein structures on almost co-Kähler manifolds. method Analyzing (m,ρ)-quasi-Einstein metrics on almost co-Kähler manifolds and studying their properties. result The paper proves that (m,ρ)-quasi-Einstein structures on almost co-Kähler manifolds are rare and have specific properties. Study characterizes Einstein manifolds in almost Ricci solitons.
problem Characterizing Einstein manifolds in almost Ricci solitons.
method Geometric dynamics and geometric analysis methods.
result Characterizes Einstein manifolds in the class of complete almost Ricci solitons.
We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely …
New Einstein metrics found in curved spaces.
problem Finding Einstein metrics in curved spaces.
method Analyzing almost-Einstein metrics to find genuine Einstein metrics.
result Negative curvature preserved in Einstein metrics.
It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-He…
Study second-Chern-Einstein metrics on 4D manifolds, finding Killing vector fields and examples.
problem Investigate second-Chern-Einstein metrics on 4D almost-Hermitian manifolds.
method Analyze compact and unimodular almost-abelian Lie algebras, use Killing vector fields and parallel non-zero Lee forms.
result Describe 4D compact second-Chern-Einstein locally conformally symplectic manifolds and classify unimodular almost-abelian Lie algebras with second-Chern-Einstein metrics.
In this article, we study almost cosymplectic manifolds admitting quasi-Einstein structures (g,V,m,λ). First we prove that an almost cosymplectic (κ,μ)-manifold is locally isomorphic to a Lie group if (g,V,m,λ) is closed and on a compact almost (κ,μ)-cosymplectic manifold there do not exist quasi-Einstein…
The paper proves rigidity results for Einstein manifolds with specific geometric constraints.
problem Understanding the rigidity of Einstein manifolds under bounded covering geometry.
method Analyzing Einstein manifolds with bounded covering geometry to prove rigidity results.
result Compact Einstein manifolds with specific geometric properties are isometric to space forms.
In this paper, we study the structure of the limit space of a sequence of almost Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the L1-sense,…
Study null geodesics on even-dimensional conformal manifolds, finding Einstein metrics and CR structures.
problem Investigate null geodesics and their geometric properties on conformal manifolds.
method Analyze the Weyl tensor and its effects on the geometry of null geodesic congruences.
result Find Einstein metrics and CR structures on the leaf space of null geodesic congruences.
The paper constructs almost para-Kähler-Einstein metrics on cotangent bundles.
problem Developing metrics on cotangent bundles associated with geometric structures.
method Using a construction involving geometric structures on a manifold M to associate an almost para-Kähler-Einstein metric on T∗M. result Explicit formulae for these metrics are derived in specific geometric cases.
Study on solitons in specific geometric manifolds, proving manifold properties and presenting examples.
problem Exploring solitons in almost coKähler and asymptotically harmonic manifolds.
method Analyzing η-Einstein and gradient η-Einstein solitons in various geometric frameworks. result Proves manifold properties and presents examples validating results.
The paper characterizes gradient solitons in specific manifold types.
problem Characterizing gradient solitons in almost Kenmotsu manifolds.
method Analyzing (m,ρ)-quasi Einstein solitons within two classes of almost Kenmotsu manifolds. result Characterized gradient (m,ρ)-quasi Einstein solitons in specific manifold types. The paper presents the Einstein connection for nonsymmetric pseudo-Riemannian manifolds.
problem The challenge is to define the Einstein connection for nonsymmetric pseudo-Riemannian manifolds.
method The approach involves using a weak almost contact structure and a linear connection with torsion.
result Explicit formulas for the Einstein connection are provided.
On asymptotically complex hyperbolic (ACH) Einstein manifolds, we consider a certain variational problem for almost complex structures compatible with the metric, for which the linearized Euler-Lagrange equation at Kähler-Einstein structures is given by the Dolbeault Laplacian acting on (0,1)-forms with values in the…
The paper presents the Einstein connection for nonsymmetric pseudo-Riemannian manifolds with specific torsion conditions.
problem Tackles the Einstein connection in nonsymmetric pseudo-Riemannian manifolds with non-degenerate skew-symmetric tensor.
method Explicitly presents the Einstein connection for nonsymmetric pseudo-Riemannian manifolds with f2-torsion condition. result Reduces to M.Prvanović's solution in the almost Hermitian case.
Study of generalized almost-Kähler-Ricci solitons and their implications.
problem Existence of first-Chern-Einstein almost-Kähler metrics on compact symplectic Fano manifolds.
method Generalization of Kähler-Ricci solitons to almost-Kähler setting, study of moment map and Lie algebra of holomorphic vector fields.
result Existence of generalized almost-Kähler-Ricci solitons as obstructions and implications for symplectic Fano manifolds.
Eta-Einstein and (κ,μ)-structures studied in dimension 3.
problem Characterizing and understanding Eta-Einstein and (κ,μ)-structures in 3D. method Analyzing closed manifolds and constructing examples.
result Almost regular Eta-Einstein structures not D-homothetic to Einstein structures exist.
The paper studies Einstein-Hilbert action on complex manifolds.
problem Deriving equations for the Einstein-Hilbert action on almost k-product manifolds. method Adapted variations of metric, deriving Euler-Lagrange equations.
result Presented a nice form of Einstein equation.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
The paper characterizes Kenmotsu metrics as almost ∗-Ricci solitons.
problem Characterizing Kenmotsu metrics as almost ∗-Ricci solitons. method Analyzing the geometry of almost contact metrics through ∗-Ricci solitons. result Kenmotsu metrics are characterized as almost ∗-Ricci solitons under specific conditions. New pseudo-Kähler Einstein spaces found with special almost complex structures.
problem Identifying new pseudo-Kähler Einstein spaces with special almost complex structures.
method Examining 4-symmetric symplectic spaces with compatible almost complex structures.
result Found pseudo-Kähler Einstein spaces with an integrable and non-integrable pair of compatible almost complex structures.
In this paper we introduce the notion of generalized quasi--Einstein manifold, that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi--Einstein manifolds. We prove that a complete generalized quasi--Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature, is locally a war…
In this article, we consider the almost Hermitian structure on TM induced by a pair of a metric and an affine connection on M. We find the conditions under which TM admits almost Kähler structures, Kähler structures and Einstein metrics, respectively. Moreover, we give two examples of Kähler-Einstein structures o…
The paper proves compactness theorems for specific types of tensors.
problem Proving compactness theorems for Riemannian manifolds with specific tensors.
method Using h-almost Ricci tensors and generalized quasi-Einstein tensors, the paper extends previous theorems. result Theorems are extended to cases where h has at most linear growth. Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.
problem Deriving critical metrics for the Einstein-Hilbert functional on compact Riemannian manifolds.
method Restricts the variational problem to an infinite-dimensional subspace.
result Derives a novel structural characterization of critical metrics.
On a 4-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of…
We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almos…
The paper characterizes Einstein metrics in Kenmotsu manifolds using specific soliton types.
problem Characterizing Einstein metrics in Kenmotsu manifolds using specific soliton types.
method Proving properties of Kenmotsu metrics as η-Ricci solitons and gradient η-Ricci solitons. result Kenmotsu metrics as η-Ricci solitons are Einstein if certain conditions are met. In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n+1)-dimensional Sasakian manifold admits a weakly Einstein metric then its scalar curvature s satisfies −6⩽s⩽6 for n=1 and $-2n(2n+1)\frac{4n^2-4n+3}{4n^2-4n-1}\leqslant s \leqslant …
Study clarifies almost Ricci-Bourguignon solitons and their properties.
problem Understanding the properties of almost Ricci-Bourguignon solitons.
method Revisit and compare with known results of Barros and Ribeiro.
result Identify conditions for compact almost RB-solitons to be trivial or have special properties.
Survey updates knowledge on homogeneous Einstein spaces.
problem Understanding homogeneous Einstein spaces.
method Building on previous surveys by Wang and Lauret.
result Current state of homogeneous Einstein spaces.
It is a well known fact that, if Σ is an Einstein hypersurface with positive scalar curvature, then it is a round sphere. We give a stable version of this result showing that if a hypersurface is almost Einstein in a Lp-sense, then it is W2,p - close to the round sphere. The result is given in a quantitat…
Unique domain found in Einstein universe, simplifying manifold classification.
problem Classifying closed conformally flat manifolds with proper development.
method Identifying and analyzing almost-homogeneous domains in the Einstein universe.
result Found a unique domain (diamond) in the Einstein universe that simplifies manifold classification.
This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection with skew symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is …
Conditions for statistical structures on manifolds derived from solitons.
problem Characterizing statistical structures on manifolds from soliton equations.
method Analyzing gradient solitons on statistical manifolds to derive conditions for statistical structures.
result Established necessary and sufficient conditions for statistical structures under various soliton types.
Constructs initial data for Einstein vacuum equations involving multiple localized gravitational sources.
problem Modeling the interaction of distant gravitational systems in general relativity.
method Time-symmetric initial data construction using gluing schemes and localized sources.
result Produces initial data sets with finite ADM mass and multiple Einstein-Rosen bridges.
The paper defines a new connection on sub-Riemannian manifolds and explores conditions for almost quasi-Sasakian manifolds to be Einstein.
problem Exploring new connections and conditions for specific types of manifolds.
method Defining a skew-symmetric connection and studying its properties on sub-Riemannian manifolds, and examining conditions for almost quasi-Sasakian manifolds to be Einstein.
result Sufficient conditions are found for an almost quasi-Sasakian manifold to be an Einstein manifold.
Einstein like (ε)-para Sasakian manifolds are introduced. For an (ε)-para Sasakian manifold to be Einstein like, a necessary and sufficient condition in terms of its curvature tensor is obtained. The scalar curvature of an Einstein like (ε)-para Sasakian manifold is obtained and it…
Study on para-Sasakian metrics and their solitons.
problem Characterizing para-Sasakian metrics with conformal η-Ricci solitons.
method Analyzing the properties of para-Sasakian metrics under conformal η-Ricci solitons.
result Para-Sasakian metrics admitting conformal η-Ricci solitons are η-Einstein.
Study on Ricci solitons and Einstein metrics in weak β-Kenmotsu manifolds.
problem Characterizing Einstein metrics in weak β-Kenmotsu manifolds.
method Adapted ∗-Ricci tensor to weak almost contact manifolds and studied its interaction with weak β-Kenmotsu structures. result New characteristics of Einstein metrics obtained.
In this paper, we prove new pinching theorems for the first eigenvalue of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for the first eigenvalue in terms of higher order mean curvatures. We show that under a suitable pinching condition, the hyp…