No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
Classifies weakly Einstein submanifolds in space forms satisfying specific equalities.
problem Characterizing submanifolds in space forms with certain geometric properties.
method Classification based on Chen's equality and semisymmetric conditions.
result Classification of weakly Einstein submanifolds in space forms.
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.
Study characterizes Einstein metrics in warped product spaces.
problem Characterizing Einstein metrics in warped product spaces.
method Local characterizations and global restatements of known results.
result Restated global characterizations of Einstein manifolds.
Study classifies Einstein spaces and warped products in weighted geometry.
problem Characterizing geometric structures of weighted Einstein spaces.
method Complete local classification of weighted Einstein spaces with harmonic Weyl tensor.
result Spaces decompose into Einstein or specific warped products.
In this article, we study Einstein Kropina metrics on Lie groups and homogeneous spaces. We give a method to construct Einstein Kropina metrics on Lie groups. As an example of this method, a family of non-Riemannian Einstein Kropina metrics on the special orthogonal group SO(n) is given. Then, we classify all left in…
Study static Einstein-Maxwell space invariant by translation.
problem Classify static Einstein-Maxwell space invariant under translation.
method Analyze static Einstein-Maxwell space conformal to pseudo-Euclidean space.
result Complete classification of static Einstein-Maxwell space invariant under translation.
Based on a well-known fact that there are no Einstein hypersurfaces in a non-flat complex space form, in this article we study the quasi-Einstein condition, which is a generalization of an Einstein metric, on the real hyersurface of a non-flat complex space form. For the real hypersurface with quasi-Einstein metric of …
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.
New formula for Lichnerowicz Laplacian on homogeneous spaces.
problem Finding new Einstein metrics on homogeneous spaces.
method Using Casimir operators to derive a new formula for the Lichnerowicz Laplacian.
result Derives many new Einstein metrics stable in the Einstein-Hilbert sense.
The study examines Einstein-Finsler spaces using Minkowskian products.
problem Characterizing Einstein-Finsler spaces through Minkowskian products.
method Proving conditions for Einstein-Finsler spaces in terms of Minkowskian products.
result If a Minkowskian product Finsler manifold is Einstein, then either the product manifold is Ricci flat or both quotient manifolds are Einstein with same scalar functions.
New proof shows perturbed non-compact Einstein spaces attract to unique global solution.
problem Proving global solutions for perturbed non-compact negative Einstein spaces.
method Developed energy estimates for a hyperbolic system of Maxwell type.
result Global unique solution for perturbed non-compact negative Einstein spaces.
Classifies smooth metric measure spaces with two weighted Einstein representatives.
problem Classifying smooth metric measure spaces with specific weighted Einstein properties.
method Local and global classification using Einstein and quasi-Einstein warped products.
result Global classification result for complete manifolds, showing specific types of manifolds.
Recently, it is proven that generalized Robertson-Walker space-times in all orthogonal subspaces of Gray's decomposition but one(unrestricted) are perfect fluid space-times. GRW space-times in the unrestricted subspace are identified by having constant scalar curvature. Generalized quasi-Einstein GRW space-times have a…
Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
Study on stability of minimal submanifolds in specific Einstein manifolds.
problem Investigating stability of minimal submanifolds in Einstein manifolds.
method Analyzing homogeneous minimal hypersurfaces in Page space and Sasaki-Einstein manifolds, computing stability operators and indices.
result Determined all homogeneous, minimal hypersurfaces and computed their stability operators and indices.
We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come f…
In this paper, we study invariant Einstein metrics on Ledger-Obata spaces Fm/diag(F). In particular, we classify invariant Einstein metrics on F4/diag(F) and estimate the number of invariant Einstein metrics on general Ledger-Obata spaces Fm/diag(F).
Local and global classifications of Einstein submanifolds in Euclidean space.
problem Classifying Einstein submanifolds in Euclidean space.
method Local and global parametric classifications with emphasis on intrinsic assumptions.
result Local and global classifications of Einstein submanifolds of codimension two.
Surveying stability and deformation of Einstein metrics.
problem Stability and deformation of Einstein metrics.
method Study of the spectrum and eigentensors of the Lichnerowicz Laplacian.
result Recent results on stability and deformation theory of Einstein metrics.
Classifies weakly Einstein hypersurfaces in spaces of constant curvature.
problem Identifying weakly Einstein hypersurfaces in spaces of constant curvature.
method Complete classification through tensor analysis and geometric properties.
result Hypersurfaces are either products of spaces of constant curvature or rotation hypersurfaces.
Study on stability of non-diagonal Einstein metrics on specific homogeneous spaces.
problem Stability analysis of non-diagonal Einstein metrics on HimesH/ΔK. method Formula for scalar curvature, study of stability with Hilbert action.
result Non-diagonal Einstein metrics on M are unstable with different coindexes. Classifies weakly Einstein curvature tensors in 4D Euclidean space.
problem Classifying algebraic curvature tensors in 4D Euclidean space.
method Algebraic formulation and geometric interpretation of weakly Einstein manifolds.
result Complete classification of non-Einstein weakly Einstein curvature tensors in dimension four.
The paper explores p-biharmonic hypersurfaces in Einstein and conformally flat spaces.
problem Characterizing p-biharmonic submanifolds in Einstein spaces.
method Analyzing properties and constructing examples of p-biharmonic hypersurfaces.
result New examples of proper p-biharmonic hypersurfaces constructed.
Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein space) which we call crooked surfaces. The conformal model of anti-de Sitter space is…
Proves stability of Einstein metrics on specific symmetric spaces.
problem Stability of Einstein metrics on symmetric spaces of compact type.
method Linear stability analysis of the Einstein-Hilbert action.
result Resolves stability problem for irreducible symmetric spaces of compact type.
Study on Einstein metrics on complex projective spaces with specific group actions.
problem Finding Einstein metrics invariant under cohomogeneity one Lie group actions.
method Analyzing Einstein equation for diagonal invariant metrics under five Takagi models.
result Nonexistence of smooth globally defined invariant Einstein metrics in four models, necessary condition in the fifth.
New Einstein manifolds split into symmetric and compact parts.
problem Understanding Einstein manifolds with unimodular isometry groups.
method Theory of polar actions, Lie-theoretic arguments, and maximum principles.
result Negative Einstein manifolds split into symmetric and compact parts.
Study on Einstein warped spaces with specific connections and curvature properties.
problem Characterizing Einstein warped spaces with quarter symmetric connections.
method Analyzing curvature, Ricci, and scalar tensors with quarter symmetric connections.
result Proves conditions under which an Einstein warped space becomes a Riemannian product space.
The paper proves Simon's conjecture for closed Einstein spaces, setting eigenvalue bounds.
problem Proving Simon's conjecture for closed Einstein spaces.
method Developed a method to prove Simon's conjecture for closed Einstein spaces.
result Proved Simon's conjecture for closed Einstein spaces, setting eigenvalue bounds.
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.
Study classifies certain Einstein 4-manifolds with twistorial properties.
problem Classifying Einstein manifolds with positive scalar curvature.
method Proving properties of Einstein four-manifolds and their twistor spaces.
result Compact Einstein four-manifolds with positive scalar curvature and specific twistorial conditions are S4 and CP2. Study Einstein-Weyl spaces from Segre quartic surfaces, finding unique geodesics and deformations.
problem Characterize Einstein-Weyl spaces associated with Segre quartic surfaces.
method Explicit construction and analysis of minitwistor spaces, focusing on singularities and geodesics.
result Found unique closed geodesics on Einstein-Weyl spaces, showing deformations and non-compactifications.
Completed classification of Einstein spaces with specific metric properties.
problem Classifying Einstein spaces with a specific type of Stackel metric.
method Invariant under three-parameter abelian group of motions, completed classification of vacuum and electrovacuum spaces.
result Complete list of metrics for Einstein spaces in privileged coordinate systems.
Classifies vacuum static spaces with harmonic curvature.
problem Classifying vacuum static spaces with harmonic curvature.
method Thorough classification through geometric analysis.
result Spaces are locally isometric to four types.
New insights into Einstein hypersurfaces in symmetric spaces.
problem Characterizing Einstein hypersurfaces in irreducible symmetric spaces.
method Analyzing properties of hypersurfaces in symmetric spaces of rank greater than 1.
result Classification of Einstein hypersurfaces in symmetric spaces of rank greater than 1.
Study the stability of Einstein metrics on homogeneous spaces.
problem Classify Einstein metrics on homogeneous spaces.
method Analyze the scalar curvature functional to understand stability.
result Identify the nature of each Einstein metric as a critical point.
A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs …
Compact Einstein manifolds with specific curvature operators are constant curvature spaces.
problem Characterizing compact Einstein manifolds with certain curvature operators.
method Bochner technique and Li's result [10]
result Compact Einstein manifolds with specific curvature operators are constant curvature spaces.
Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
Study Einstein metrics on HimesH/ΔK spaces.
problem Existence and classification of invariant Einstein metrics on HimesH/ΔK. method Investigate HimesH-invariant Einstein metrics on M=HimesH/ΔK. result Find unstable Einstein metrics on M for many spaces H/K. New Einstein RCD spaces found with cone singularities.
problem Existence of Einstein RCD spaces with cone singularities.
method Characterization of RCD spaces and cone singularity analysis.
result Existence of smooth non-compact 4-manifolds with ALE Ricci-flat RCD(0,4) metrics.
New findings on Kähler-Einstein currents dominating Kähler forms on compact spaces.
problem Understanding Kähler-Einstein currents on singular spaces.
method Analyzing currents' properties and applying them to compact spaces with log terminal singularities.
result Kähler-Einstein currents dominate Kähler forms on compact spaces with log terminal singularities.
Solves classical problem with Kähler-Einstein metrics in complex projective spaces.
problem Classical problem of non-isometric bidimensional Kähler-Einstein submanifolds.
method Listed complete non-isometric bidimensional rotation invariant Kähler-Einstein submanifolds.
result Solves the classical problem in the specified case.
Characterizes warping functions in Einstein Poisson warped spaces.
problem Existence and nonexistence of warping functions with constant scalar curvature.
method Analyzes various dimensions of base space and constant scalar curvature conditions.
result Characterizes warping functions for different dimensions of base space.
Study classifies Einstein-Yang-Mills spaces in 4D symmetric spaces.
problem Classifying Lorentzian symmetric spaces with Einstein-Yang-Mills properties.
method Classification based on invariant metric connections and diagonal metrics.
result Four-dimensional symmetric spaces with nontrivial isotropy groups are classified.
No Einstein metrics found on certain double disk bundles.
problem Existence of Einstein metrics on specific manifolds.
method Phase space barrier argument to show non-existence.
result Proves non-existence of cohomogeneity one Einstein metrics.
This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of piecewise linear spaces we define and discuss (normalized) Ricci flows. …