Authors classify 3D m-quasi Einstein manifolds with degenerate Ricci tensor.
problem Classifying 3D m-quasi Einstein manifolds with degenerate Ricci tensor.
method Using Codazzi tensor and geometric properties of the tensor to analyze m-quasi Einstein equation.
result Explicit description of local and complete metrics and potential functions.
The paper studies m-quasi Einstein manifolds with convex potential and finds constant scalar curvature.
problem Investigating m-quasi Einstein manifolds with a convex potential function. method Analyzing integral conditions and properties of the potential vector field.
result An m-quasi Einstein manifold with a convex potential function has constant scalar curvature. The paper analyzes symmetry groups of a specific type of manifold.
problem Understanding the symmetry groups of generalized m-quasi-Einstein manifolds. method Analyzing a n-dimensional generalized m-quasi-Einstein manifold conformal to a pseudo-Euclidean space. result Proves the most general symmetry group of maximal dimension and shows no different low-dimensional invariants.
We call a metric m-quasi-Einstein if RicXm, which replaces a gradient of a smooth function f by a vector field X in m-Bakry-Emery Ricci tensor, is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant met…
The paper classifies quasi-Einstein 3-manifolds and their properties.
problem Classifying compact locally homogeneous non-gradient quasi-Einstein 3-manifolds.
method Analyzing quotient spaces of Lie groups and using properties of quasi-Einstein metrics.
result Identifies conditions for the existence of nontrivial quasi-Einstein metrics.
We call a metric m-quasi-Einstein if RicXm (a modification of the m-Bakry-Emery Ricci tensor in terms of a suitable vector field X) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and…
Study on rigidity of special Riemannian manifolds.
problem Rigidity properties of generalized m-quasi-Einstein manifolds of Yamabe-type. method Investigation of rigidity properties for the potential vector field in compact and non-compact settings.
result The potential vector field either vanishes identically or becomes a non-trivial Killing vector field under certain assumptions.
Killing fields on compact m-quasi-Einstein manifolds are shown under specific curvature conditions.
problem Characterizing Killing fields on compact m-quasi-Einstein manifolds.
method Extending a result by Bahuaud-Gunasekaran-Kunduri-Woolgar, the approach involves proving the existence of Killing fields under certain curvature conditions.
result A sufficient condition for a compact, non-gradient m-quasi-Einstein metric to admit a Killing field is provided, extending the original result to the m = -2 case.
Study on noncompact steady quasi-Einstein manifolds with specific tensor conditions.
problem Classifying noncompact steady quasi-Einstein manifolds with vanishing Weyl tensor condition.
method Analyzing manifolds with nonnegative Ricci curvature and zero radial Weyl curvature under fourth-order divergence-free Weyl tensor condition.
result Proves that such manifolds must be a warped product with (n−1)−dimensional Einstein fiber. The paper proves a spin manifold's 4D quasi-Einstein satisfies Hitchin-Thorpe inequality.
problem Proving a specific inequality for a class of 4D manifolds.
method Analyzing properties of gradient m-quasi-Einstein manifolds, focusing on spin structures. result Compact 4D spin gradient m-quasi-Einstein manifolds satisfy the Hitchin-Thorpe Inequality when m≥1. Study on compact Quasi-Einstein manifolds yields diameter estimates and Hitchin-Thorpe inequality conditions.
problem Estimating diameters and verifying Hitchin-Thorpe inequality for compact Quasi-Einstein manifolds.
method Derive geometric estimates relating potential function oscillation to manifold diameter; derive lower bounds for diameter.
result Diameter conditions ensure compact Quasi-Einstein manifolds satisfy Hitchin-Thorpe inequality in dimension four.
Study rigidifies non-compact manifolds with specific curvature conditions.
problem Analyzing non-compact generalized m-quasi-Einstein manifolds with constant scalar curvature and soliton function.
method Introduced a weighted function and proved its subharmonicity to derive rigidity results.
result Proves manifolds are Euclidean under specific conditions, with constant μ essential.
Extends Euler class formula to general connections with metric.
problem Formula for Euler class with metric connection.
method Rewriting formula for general connections with metric.
result Rewritten Gauss-Bonnet theorem in dimension two.
The study extends and generalizes a result about quasi Einstein manifolds, proving conditions for Killing vector fields.
problem Characterizing conditions for Killing vector fields in quasi Einstein manifolds.
method Extending and generalizing Cochran's result, proving conditions for Killing vector fields under specific integrals and conformal conditions.
result Conditions for Killing vector fields in quasi Einstein manifolds, including integral identities and global isometry to spheres.
The study characterizes spacetimes with specific solitons in f(R)-gravity.
problem Characterizing spacetimes with specific solitons in f(R)-gravity. method Analyzing η-Ricci solitons, gradient η-Ricci solitons, gradient Einstein Solitons, and gradient m-quasi Einstein solitons in perfect fluid spacetimes obeying f(R)-gravity. result Established conditions for the behavior of η-Ricci solitons and derived significant theorems about dark matter. This study classifies noncompact quasi-Einstein manifolds conformal to Euclidean spaces.
problem Investigating nontrivial quasi-Einstein manifolds globally conformal to Euclidean spaces.
method Considering manifolds with invariant conformal factors and potential functions under an (n−1)-dimensional translation group. result Complete classification of manifolds when λ=0 and m≥1 or m=2−n. The study classifies quasi-Einstein manifolds with constant scalar curvature.
problem Characterizing quasi-Einstein manifolds with specific curvature properties.
method Classification and construction of examples of quasi-Einstein manifolds.
result Complete classification of quasi-Einstein manifolds with constant scalar curvature.
New quasi-Einstein metrics found on a sphere.
problem Finding quasi-Einstein metrics on a sphere.
method Constructing axi-symmetric non-gradient m-quasi-Einstein structures using hypergeometric functions. result Found new regular metrics on a two-sphere, including the extreme Kerr black hole horizon.
The paper explores generalized quasi-Einstein manifolds and their properties.
problem Investigating properties of generalized quasi-Einstein manifolds under specific conditions.
method Analyzing natural conditions on potential vector fields and deriving consequences.
result The potential vector field is shown to be Killing under suitable integral assumptions.
Study of 3D degenerate Riemannian manifolds satisfying specific geometric equations.
problem Characterizing 3D degenerate Riemannian manifolds with solutions to a geometric equation.
method Developed a general approach to solve the equation \(
abla df = \psi Rc + \varphi g\), specifying the metric \(g\) under certain conditions.
result Explicitly described the metric \(g\) and potential function \(f\) for various classes of 3D degenerate spaces.
The paper finds new Ricci solitons from Hopf fibrations.
problem Finding new steady and expanding Ricci solitons.
method Analyzing cohomogeneity one Ricci solitons with specific isotropy representations.
result Existence of parameter families of non-homothetic complete steady and expanding Ricci solitons.
In this paper we take the perspective introduced by Case-Shu-Wei of studying warped product Einstein metrics through the equation for the Ricci curvature of the base space. They call this equation on the base the m-Quasi Einstein equation, but we will also call it the (λ,n+m)-Einstein equation. In this paper we ext…
New bounds on black hole topology without symmetry assumptions.
problem Understanding the topology of extreme black holes without symmetry constraints.
method Using near-horizon geometries and m-quasi Einstein metrics, combined with generalizations of the splitting theorem. result Refined classifications of black hole topologies without symmetry assumptions.
The study examines perfect fluid spacetimes and their properties.
problem Characterizing properties of perfect fluid spacetimes with concircular vector fields.
method Analyzing the conformal curvature tensor, state equation, and solitons in perfect fluid spacetimes.
result Perfect fluid spacetimes with concircular vector fields have specific properties related to the state equation and solitons.
The study investigates properties of a specific Riemannian manifold with a semi-symmetric non-metric connection.
problem Characterizing properties of a Riemannian manifold with a semi-symmetric non-metric connection.
method Construction of a non-trivial example, proving manifold properties based on the metric being a gradient soliton or Yamabe soliton.
result A manifold with a semi-symmetric non-metric connection and gradient Ricci/Yamabe soliton is of constant curvature.
The paper classifies warped product almost Ricci solitons.
problem Understanding warped product almost Ricci solitons.
method Analyzing Ricci-Hessian type manifolds and considering two complementary cases.
result The vector field \(
abla\lambda\) belongs to the \(C^\infty(\Bbb{M})\)-module generated by \(
abla f\) and \(
abla\varphi\) in the first case.
Let G be a compact, connected Lie group, acting smoothly on a manifold M. Goresky-Kottwitz-MacPherson described a small Cartan model for the equivariant cohomology of M, quasi-isomorphic to the standard Cartan complex of equivariant differential forms. In this paper, we construct an explicit cochain map from the small …
With any non necessarily orientable unpunctured marked surface (S,M) we associate a commutative algebra, called quasi-cluster algebra, equipped with a distinguished set of generators, called quasi-cluster variables, in bijection with the set of arcs and one-sided simple closed curves in (S,M). Quasi-cluster variables a…
An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals…
Bismut Einstein metrics on complex manifolds are Kähler Einstein or Bismut Ricci flat.
problem Characterizing Bismut Einstein metrics on compact complex manifolds.
method Observing the (2,0)-part of Bismut Ricci form and using it to prove properties of the metrics.
result Bismut Einstein metrics with non-zero Einstein constant are Kähler Einstein, and those with zero are Bismut Ricci flat.
Study on Einstein deformations of negative Kähler Einstein metrics.
problem Understanding Einstein deformations of Kähler Einstein metrics.
method Relate second order Einstein deformation theory to complex geometry, gauge normalise, and use Taylor expansion.
result Taylor expansion to order two of an Einstein deformation is determined by h12 and the divergence of the Kodaira-Spencer bracket. Homogeneous Einstein metrics on Euclidean spaces are shown to be Einstein solvmanifolds.
problem Characterizing homogeneous Einstein metrics on Euclidean spaces.
method Using periodic, integrally minimal foliations and geometric flow induced by the orbit-Einstein condition.
result Homogeneous Einstein metrics on Euclidean spaces are proven to be Einstein solvmanifolds.
The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.
problem Exploring new types of manifolds in general relativity.
method Generalization of existing manifolds and construction of specific examples.
result Existence and properties of extended quasi-Einstein manifolds with solitons.
Study shows Einstein structures on 4-manifolds are rigid.
problem Rigidity of Einstein structures in four dimensions.
method Examined deformations of the round four-sphere and analyzed self-dual structure of Einstein manifolds.
result Any deviation from the standard metric of the round four-sphere breaks the Einstein condition.
New types of Einstein manifolds discovered from homogeneous surfaces.
problem Understanding quasi-Einstein equations on homogeneous surfaces.
method Modified Riemannian extension and warped product construction.
result Discovery of new Einstein manifolds.
The study explores Einstein-Weyl structures on specific types of manifolds.
problem Investigating properties of Einstein-Weyl structures on almost cosymplectic manifolds.
method Analyzing conditions for Einstein-Weyl structures on (κ,μ)-manifolds, three-dimensional compact manifolds, and K-cosymplectic manifolds. result Conditions for manifolds to be Einstein, cosymplectic, or Ricc-flat.
New framework detects Einstein metrics using harmonic maps.
problem Local deformation of compact cohomogeneity-one Einstein metrics.
method Intrinsic reformulation of Einstein boundary-value problem combined with equivariant harmonic maps.
result Einstein Detection Principle for local deformation theory.
This paper proves certain quasi-Einstein manifolds are rigid under Ricci flow.
problem Understanding the behavior of quasi-Einstein metrics under Ricci flow.
method Employing a curvature evolution identity associated with Ricci flow.
result Certain closed quasi-Einstein manifolds are rigid under Ricci flow.
Study Einstein warped products with Einstein base and fiber.
problem Characterize Einstein warped products with Einstein base and fiber.
method Investigate necessary and sufficient conditions for a warped product to be Einstein.
result Explicitly determine the warping function when the base is hyperbolic space.
Classifies non-Einstein solutions to Einstein--Maxwell equations on 4D Lie algs.
problem Finding non-Einstein solutions to Einstein--Maxwell equations on 4D Lie algebras.
method Classification of left-invariant solutions up to automorphisms.
result Classification of all left-invariant non-Einstein solutions.
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 paper shows that certain Einstein orbifolds cannot be limits of smooth Einstein metrics.
problem Understanding the limits of smooth Einstein metrics on compact Einstein orbifolds.
method Analyzing sequences of compact Einstein manifolds and their limits, providing an explicit obstruction for certain orbifolds.
result Explicit obstruction for negative Einstein orbifolds appearing as limits of compact Einstein manifolds, which does not vanish for hyperbolic orbifolds.
New connections on 5-manifolds linked to Sasaki-Einstein structures.
problem Finding connections on 5-manifolds with specific properties.
method Using skew-symmetric torsion and Einstein metricity conditions.
result Existence of connections on 5-manifolds is equivalent to the existence of Sasaki-Einstein 5-manifolds.
The study finds quasi-Einstein metrics on sphere bundles.
problem Finding quasi-Einstein metrics on specific types of manifolds.
method Adapting Hall's work, the study explores quasi-Einstein metrics on sphere bundles over Fano Kaehler-Einstein manifolds and their blow-downs.
result The discovery of quasi-Einstein metrics on sphere bundles.
The study explores Einstein Kropina metrics on Lie groups and homogeneous spaces.
problem Investigating Einstein Kropina metrics on Lie groups and homogeneous spaces.
method Constructing Einstein Kropina metrics on Lie groups and homogeneous spaces using specific procedures.
result Classification and construction of Einstein Kropina metrics on various Lie groups and homogeneous spaces.
Weakly Einstein Kähler surfaces are characterized and classified.
problem Characterizing and classifying weakly Einstein Kähler surfaces.
method Several conditions and constructions to characterize and classify weakly Einstein Kähler surfaces.
result Classification of weakly Einstein Kähler surfaces with specific properties and construction of new examples.
Kaehler-Einstein metrics on orbifolds derived from Einstein sequences.
problem Desingularizing Einstein orbifolds with Kaehler-Einstein metrics.
method Analyzing sequences of smooth compact Einstein 4-manifolds converging to orbifolds.
result The limit orbifold is Kaehler-Einstein and one of the classified orbifold limits.
The study proves conditions for quasi-Einstein manifolds with specific structures to be Einstein.
problem Conditions for quasi-Einstein manifolds to have Einstein structures.
method Proved conditions for quasi-Einstein semi-Riemannian warped products to have Einstein fibers.
result Found conditions for quasi-Einstein manifolds with specific structures to be Einstein.