Einstein Metrics Research Articles

Generalized teleparallel de Sitter geometries

Theories of gravity based on teleparallel geometries are characterized by the torsion, which is a function of the coframe, derivatives of the coframe, and a zero curvature and metric compatible spin-connection. The appropriate notion of a symmetry in a teleparallel geometry is that of an affine symmetry. Due to the importance of the de Sitter geometry and Einstein spaces within General Relativity, we shall describe teleparallel de Sitter geometries and discuss their possible generalizations. In particular, we shall analyse a class of Einstein teleparallel geometries which have a 4-dimensional Lie algebra of affine symmetries, and display two one-parameter families of explicit exact solutions.

On Semi-Riemannian Manifolds Satisfying Some Generalized Einstein Metric Conditions

The derivation-commutator $R \cdot C - C \cdot R$ of a semi-Riemannian manifold $(M,g)$, $\dim M \geq 4$, formed by its Riemann-Christoffel curvature tensor $R$ and the Weyl conformal curvature tensor $C$, under some assumptions, can be expressed as a linear combination of $(0,6)$-Tachibana tensors $Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$ a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.

Six-dimensional one-loop divergences in quantum gravity from the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 spinning particle

In this work, we investigate the computation of the counterterms necessary for the renormalization of the one-loop effective action of quantum gravity using both the worldline formalism and the heat kernel method. Our primary contribution is the determination of the Seleey-DeWitt coefficient a3(D) for perturbative quantum gravity with a cosmological constant, which we evaluate on Einstein manifolds of arbitrary D dimensions. This coefficient characterizes quantum gravity in a gauge-invariant manner due to the on-shell condition of the background on which the graviton propagates. Previously, this coefficient was not fully known in the literature. We employ the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 spinning particle model recently proposed to describe the graviton in first quantization and then use the heat kernel method to cross-check the correctness of our calculations. Finally, we restrict to six dimensions, where the coefficient corresponds to the logarithmic divergences of the effective action, and compare our results with those available in the literature.

Convergence of the Hesse–Koszul flow on compact Hessian manifolds

We study the long time behavior of the Hesse–Koszul flow on compact Hessian manifolds. When the first affine Chern class is negative, we prove that the flow converges to the unique Hesse–Einstein metric. We also derive a convergence result for a twisted Hesse–Koszul flow on any compact Hessian manifold. These results give alternative proofs for the existence of the unique Hesse–Einstein metric by Cheng–Yau and Caffarelli–Viaclovsky as well as the real Calabi theorem by Cheng–Yau, Delanoë and Caffarelli–Viaclovsky.

Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds

Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds

Degenerations of negative Kähler–Einstein surfaces

AbstractEvery compact Kähler manifold with negative first Chern class admits a unique metric such that . Understanding how families of these metrics degenerate gives insight into their geometry and is important for understanding the compactification of the moduli space of negative Kähler–Einstein metrics. I study a special class of such families in complex dimension two. Following the work of Sun and Zhang in the Calabi–Yau case [2019, arXiv:1906.03368], I construct a Kähler–Einstein neck region interpolating between canonical metrics on components of the central fiber. This provides a model for the limiting geometry of metrics in the family.

Conelike radiant structures

Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structures with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein–Weyl structures over Kähler–Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for special statistical manifolds is given that generalizes the Einstein–Weyl equations and encompasses these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.

Tangent Bundles Endowed with Quarter-Symmetric Non-Metric Connection (QSNMC) in a Lorentzian Para-Sasakian Manifold

The purpose of the present paper is to study the complete lifts of a QSNMC from an LP-Sasakian manifold to its tangent bundle. The lifts of the curvature tensor, Ricci tensor, projective Ricci tensor, and lifts of Einstein manifold endowed with QSNMC in an LP-Sasakian manifold to its tangent bundle are investigated. Necessary and sufficient conditions for the lifts of the Ricci tensor to be symmetric and skew-symmetric and the lifts of the projective Ricci tensor to be skew-symmetric in the tangent bundle are given. An example of complete lifts of four-dimensional LP-Sasakian manifolds in the tangent bundle is shown.

Null hypersurfaces in 4-manifolds endowed with a product structure

AbstractIn a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric, and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the existence of null hypersurface with equal nontrivial principal curvatures, and finally, we give a necessary condition on the ambient scalar curvature, for the existence of null (non-minimal) hypersurfaces that are of constant mean curvature.

Modified conformal extensions

We present a geometric construction and characterization of 2n-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson–Walker metric construction for n-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive a complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman–Graham ambient metric and show the vanishing of the Q-curvature.

RICCI SOLITONS ON Α-PARA KENMOTSU MANIFOLDS WITH SEMI SYMMETRIC METRIC CONNECTION

In this paper we introduce notion of Ricci solitons in -para Kenmotsu manifold with semi -symmetric metric connection. We have found the relations between curvature tensor, Ricci tensors and scalar curvature of -para Kenmotsu manifold with semi-symmetic metric connection.We have proved that 3-dimensional -para Kenmotsu manifold with semi -symmetric metric connection is an -Einstein manifold and the Ricci soliton defined on this manifold is named expanding and steady with respect to the value of constant.It is proved that Conharmonically flat -para Kenmotsu manifold with semi-symmetric metric connection is -Einstein manifold.

Homogeneous Einstein manifolds

Homogeneous Einstein manifolds

Tensor Decompositions and Their Properties

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ=1,2,3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

Results of Hyperbolic Ricci Solitons

We obtain some properties of a hyperbolic Ricci soliton with certain types of potential vector fields, and we point out some conditions when it reduces to a trivial Ricci soliton. We also study those soliton submanifolds whose vector fields are the tangential components of a concurrent vector field on the ambient manifold, and in particular, we show that a totally umbilical hyperbolic Ricci soliton is an Einstein manifold. We prove that if the hyperbolic Ricci soliton hypersurface of a Riemannian manifold of constant curvature and endowed with a concurrent vector field has a parallel shape operator, then it is a metallic-shaped hypersurface, and we determine some conditions for it to be minimal. Moreover, we show that it is also a pseudosymmetric hypersurface.

A quantization proof of the uniform Yau–Tian–Donaldson conjecture

Using quantization techniques, we show that the \delta -invariant of Fujita–Odaka coincides with the optimal exponent in a certain Moser–Trudinger type inequality. Consequently, we obtain a uniform Yau–Tian–Donaldson theorem for the existence of twisted Kähler–Einstein metrics with arbitrary polarizations. Our approach mainly uses pluripotential theory, which does not involve Cheeger–Colding–Tian theory or the non-Archimedean language. A new computable criterion for the existence of constant scalar curvature Kähler metrics is also given.

Integrability of Einstein deformations and desingularizations

AbstractWe study the question of the integrability of Einstein deformations and relate it to the question of the desingularization of Einstein metrics. Our main application is a negative answer to the long‐standing question of whether or not every Einstein 4‐orbifold (which is an Einstein metric space in a synthetic sense) is limit of smooth Einstein 4‐manifolds. We more precisely show that spherical and hyperbolic 4‐orbifolds with the simplest singularities cannot be Gromov‐Hausdorff limits of smooth Einstein 4‐metrics without relying on previous integrability assumptions. For this, we analyze the integrability of deformations of Ricci‐flat ALE metrics through variations of Schoen's Pohozaev identity. Inspired by Taub's conserved quantity in General Relativity, we also introduce conserved integral quantities based on the symmetries of Einstein metrics. These quantities are obstructions to the integrability of infinitesimal Einstein deformations “closing up” inside a hypersurface – even with change of topology. We show that many previously identified obstructions to the desingularization of Einstein 4‐metrics are equivalent to these quantities on Ricci‐flat cones. In particular, all of the obstructions to desingularizations bubbling off Eguchi‐Hanson metrics are recovered. This lets us further interpret the obstructions to the desingularization of Einstein metrics as a defect of integrability.

Eigenvalue estimates for 3-Sasaki structures

Abstract We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving the Lichnerowicz–Obata-type estimates by Ivanov, Petkov and Vassilev (2013, 2014). The limiting eigenspace is fully described in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz–Matsushima estimate for Kähler–Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun (2013, 2017), in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.

Orbifold Kähler–Einstein metrics on projective toric varieties

AbstractIn this short note, we investigate the existence of orbifold Kähler–Einstein metrics on toric varieties. In particular, we show that every ‐factorial normal projective toric variety allows an orbifold Kähler–Einstein metric. Moreover, we characterize ‐stability of ‐factorial toric pairs of Picard number one in terms of the log Cox ring and the universal orbifold cover.

Renormalized Area for Minimal Hypersurfaces of 5D Poincaré–Einstein Spaces

Renormalized Area for Minimal Hypersurfaces of 5D Poincaré–Einstein Spaces

Concircularly semi-symmetric metric connection

Results on a concircularly semi-symmetric metric connection on a Riemannian manifold are presented. Six linearly independent curvature tensors with respect to this non-symmetric linear connection are studied, and the tensors coincident with the Weyl projective curvature tensor and the concircular curvature tensor of the Levi-Civita connection are determined. Transformations of connections when the Riemannian curvature tensor, Weyl projective curvature tensor and the concircular curvature tensor of the Levi-Civita connection are invariant are also observed. Moreover, new conditions for the Einstein manifold are derived.