The paper studies modified Einstein tensors and their positivity properties on compact manifolds.
problem Analyzing the positivity of modified Einstein tensors and their implications on compact manifolds.
method Investigates modified Einstein tensors defined as $\Eink :=\Scal \, g -k\Ric$ for 0<k<n and studies their positivity properties. result Defines a smooth invariant $\cEin(M)$ measuring how far a manifold is from admitting an Einstein metric with positive scalar curvature.
In this paper, we prove a far-from-CMC result similar to the ones obtained by Holst, Nagy, Tsogtgerel and Maxwell for the conformal Einstein-scalar field constraint equations on compact Riemannian manifolds with positive (modified) Yamabe invariant.
Researchers solve metric curvature equations on manifolds with boundary.
problem Finding complete conformal metrics with specific curvature functions.
method Revealed algebraic structure of fully nonlinear equations; used topological obstructions.
result Solved a class of fully nonlinear equations for conformal metrics.
Scalar-tensor gravitation theories, such as the Brans-Dicke family of theories, are commonly partly described by a modified Einstein equation in which the Ricci tensor is replaced by the Bakry-Émery-Ricci tensor of a Lorentzian metric and scalar field. In physics this formulation is sometimes referred to as the "Jordan…
Study new Einstein-like metrics and their properties.
problem Characterize a new class of quasi-Einstein metrics.
method Investigate modified Ricci solitons and their relationships.
result Prove rigidity of standard spheres under specific conditions.
Study on type-D Ricci-flat metrics with Killing spinors and Killing vectors.
problem Characterizing and solving metrics with specific properties.
method Using Killing spinors and Killing vectors, rederive results via Toda field equations and axisymmetric solutions.
result Field equations linearize for certain metrics, including axisymmetric solutions.
Study on gradient ρ-Einstein solitons with radially nonnegative Bach tensor.
problem Characterizing gradient ρ-Einstein solitons with specific tensor properties.
method Analyzing the properties of Bach tensor and using local warping to classify solitons.
result Gradient ρ-Einstein solitons with radially nonnegative Bach tensor are locally warped products of an interval and an Einstein manifold.
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.
Study classifies 3D Einstein manifolds with cyclic Ricci tensor.
problem Classifying Einstein manifolds with specific tensor properties.
method Derived integral formula involving tensor D for classification.
result Obtained rigidity results for 3D manifolds.
On a conformal manifold, it is well known that parallel sections of the standard tractor bundle with non-vanishing scale are in 1-1 correspondence with solutions of the conformal Einstein equation. In 2 dimensions conformal geometry carries no local information but one can remedy this by equipping the surface with a Mö…
New stability and isolation results for Einstein manifolds.
problem Stability and isolation of Einstein manifolds.
method Conditions on Weyl tensor for AH and ALE manifolds, Bochner tensor for Kähler and Sasaki manifolds.
result Established new stability criteria and isolation results for various types of Einstein manifolds.
We propose further conformal parametrizations for initial data in some modified Einstein gravity theories. Some of them give rise to conformally covariant systems.
Proves well-posedness for Einstein equations with totally geodesic timelike boundary condition.
problem Initial boundary value problem for Einstein equations with specific geometric boundary condition.
method ADM system, parallelly propagated orthonormal frame, modified evolution equations, hyperbolic systems, constraints propagation.
result First well-posedness result for Einstein equations with totally geodesic timelike boundary condition.
Survey on manifolds satisfying generalized Einstein conditions.
problem Characterizing semi-Riemannian manifolds under specific curvature conditions.
method Analyzing the difference tensor R.C-C.R expressed as linear combinations of Tachibana tensors.
result Recent results on manifolds and submanifolds satisfying generalized Einstein conditions.
Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.
problem Defining metrics and Einstein tensors on Riemannian manifolds.
method Defines bilinear functionals of vector fields and differential forms, generalizing to non-commutative geometry.
result Proves the vanishing of the Einstein functional for the conformally rescaled geometry of the noncommutative two-torus.
Paper extends curvature estimates to new tensor types.
problem Mean curvature and volume comparison estimates for integral generalized quasi-Einstein tensors.
method Extends existing comparison results to new tensor types.
result Global diameter estimates derived from comparison results.
The paper examines properties of W-curvature tensor in relativistic space-times.
problem Investigating the properties and implications of the W-curvature tensor in relativistic space-times. method Analyzing the semi-symmetry and divergence properties of the energy-momentum tensor in relation to the W-curvature tensor. result Space-times with specific properties of the W-curvature tensor are classified as Einstein or Codazzi type. Study curvature properties in special manifolds using specific tensors.
problem Investigate curvature conditions in 2-quasi-Einstein manifolds.
method Analyze Riemann-Christoffel curvature tensor and its linear combinations with Ricci tensor.
result Satisfy pseudosymmetry type curvature conditions in certain manifolds.
The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner type formulas for the Weyl tensor on a four dimensional Einstein manifold. In particular, we prove a second Bochner type formula which, formally, extends…
Paper classifies Einstein-type manifolds with parallel Ricci tensor.
problem Classifying Einstein-type manifolds with specific curvature properties.
method Deduced Bochner-type identity and used it to show rigidity results.
result Found conditions for classifying Einstein-type manifolds with parallel Ricci tensor.
The paper computes metrics and Einstein tensors on even-dimensional manifolds.
problem Computing metrics and Einstein tensors on even-dimensional Riemannian manifolds.
method Development of spectral Einstein functionals and equivariant Bismut Laplacian.
result Explicit computation of the equivariant noncommutative residue density.
The holographic duality can be extended to include quantum theories with broken coordinate invariance leading to the appearance of the gravitational anomalies. On the gravity side one adds the gravitational Chern-Simons term to the bulk action which gauge invariance is only up to the boundary terms. We analyze in detai…
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.
We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einst…
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 stability of Einstein manifolds with boundary.
problem Stability of Einstein manifolds with geometric boundary conditions.
method Using Ricci flow and calculus of variations, analyze stability with respect to the Einstein-Hilbert action.
result Introduce a new subspace of tensors (TVg tensors) for stability condition due to boundary constraints.
Curvature tensors can always be matched to a metric tensor under certain conditions.
problem Sectionally positive curvature tensors and their relationship to metric tensors.
method Existence and uniqueness of a metric tensor gab such that Rabcdgbd=gacλ. result A metric tensor gab can be found for sectionally positive curvature tensors, and it is unique up to a constant factor. Defines a new tensor related to special geometric spaces.
problem Understanding new curvature tensors in semi-Riemannian geometry.
method Defines a generalized curvature tensor using specific operations.
result The tensor is connected to quasi-Einstein, Roter, and Roter-type spaces.
New currents derived from Killing-Yano tensors for gravity.
problem Finding new conserved currents in gravity.
method Using relations involving Riemann, Ricci, and Einstein tensors to introduce novel conserved currents.
result New currents derived from Killing-Yano tensors and their implications for conserved charges.
Study on generalized quasi-Einstein structures in contact geometry.
problem Characterizing and understanding generalized quasi-Einstein structures in contact geometry.
method Investigation of properties, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifolds.
result Normal metric contact pair manifolds with generalized quasi-constant curvature are generalized quasi-Einstein manifolds.
Optimal pinching results on Einstein manifolds with positive Yamabe invariant.
problem Understanding the rigidity of Einstein manifolds with positive Yamabe invariant.
method Optimal pinching results and bounds on scalar curvature and Weyl tensor norms.
result Improved bounds on the Yamabe invariant and scalar curvature.
Study on 3D Lie groups finds all generalized Einstein metrics.
problem Classifying generalized Einstein metrics on 3D Lie groups.
method Developed theory of left-invariant generalized pseudo-Riemannian metrics, computed Ricci tensor, determined all metrics.
result Determined all generalized Einstein metrics on three-dimensional Lie groups.
The problem of characterizing conformally Einstein manifolds by tensorial conditions has been tackled recently in papers by M. Listing, and in work by A. R. Gover and P. Nurowski. Their results apply to metrics satisfying a "non-degeneracy" condition on the Weyl tensor \W. We investigate the geometry of the foliations …
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.
Proves Kähler-Einstein property for certain Einstein 4-manifolds.
problem Characterizing Einstein 4-manifolds with specific curvature conditions.
method Analyzes self-dual Weyl tensor and scalar curvature conditions.
result Einstein 4-manifolds are either anti-self-dual or Kähler-Einstein under certain conditions.
We study the affine quasi-Einstein Equation for homogeneous surfaces. This gives rise through the modified Riemannian extension to new half conformally flat generalized quasi-Einstein neutral signature (2,2) manifolds, to conformally Einstein manifolds and also to new Einstein manifolds through a warped product const…
Certain curvature conditions for stability of Einstein manifolds with respect to the Einstein-Hilbert action are given. These conditions are given in terms of quantities involving the Weyl tensor and the Bochner tensor. In dimension six, a stability criterion involving the Euler characteristic is given.
The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
problem Analyzing modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
method Proving Laplacian comparison theorems using modified m-Bakry-Emery Ricci tensors under m≤1.
result Optimal conditions for modified m-Bakry-Emery Ricci tensors under m≤1 are derived.
The paper approximates Einstein tensor using finite elements.
problem Approximating Einstein tensor for piecewise polynomial metrics.
method Finite element method applied to Riemannian metrics.
result Convergence rate of O(hr+1) in H−2(Ω)-norm. In this paper, we prove some compactness theorems of Myers, Ambrose, and Galloway for complete Riemannian manifold in the concept of h-almost Ricci tensors and generalized quasi-Einstein tensors. Also, we extend the previous theorems when h has at most linear growth in the distance function.
In this article we give a classification of three dimensional m-quasi Einstein manifolds with two distinct Ricci-eigen values. Our study provides explicit description of local and complete metrics and potential functions. We also describe the associated warped product Einstein manifolds in detail. For the proof we pres…
The two-jet of the curvature tensor at some point of a pseudo-Riemannian manifold is called Einstein if the Ricci tensor is a multiple of the metric tensor at the given point and additionally its first two covariant derivatives vanish there. Following the Jet Isomorphism Theorem of pseudo-Riemannian geometry, we derive…
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
The aim of this paper is to study complete (noncompact) steady m-quasi-Einstein manifolds satisfying a fourth-order vanishing condition on the Weyl tensor. In this case, we are able to prove that a steady m-quasi-Einstein manifold (m>1) on a simply connected n-dimensional manifold (Mn,g), (n≥4), with …
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…
The paper classifies compact quasi-Einstein manifolds with boundary.
problem Classifying compact quasi-Einstein manifolds with boundary.
method Analyzing manifolds with nonnegative sectional curvature and zero radial Weyl tensor.
result Classification of quasi-Einstein manifolds, including standard hemisphere and new examples.
According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the "hyperbolic" toric Kähler-Einstein equation eΦ=detD2Φ on proper convex cones. We…
The difference tensor C.R - R.C of Einstein manifolds, some quasi-Einstein manifolds and Roter type manifolds, of dimension n > 3, satisfy the following curvature condition: (A) C.R - R.C = Q(S,C) - (k /(n-1)) Q(g,C). We investigate hypersurfaces M in space forms N satisfying (A). The main result states that if the ten…