Traceless Ricci Tensor Research Articles

On minimal Lagrangian submanifolds in complex space forms with semi-parallel second fundamental form

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension [Formula: see text] in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases [Formula: see text]. Second, supposing further that the scalar curvature is constant for [Formula: see text], we also give an answer to the problem by applying the classification theorem of [F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012) 79–115]. Finally, for such Lagrangian submanifolds in the above problem with [Formula: see text], we establish an inequality in terms of the traceless Ricci tensor, the squared norm of the second fundamental form and the scalar curvature. Moreover, this inequality is optimal in the sense that all the submanifolds attaining the equality are completely determined.

Newman-Penrose formalism in quadratic gravity

The quadratic gravity constraints are reformulated in terms of the Newman-Penrose-like quantities. In such a frame language, the field equations represent a linear algebraic system for the traceless Ricci tensor components. In principle, a procedure for the combination of the Ricci components with standard geometric identities can be applied in a similar way as in the case of general relativity. These results could serve in various subsequent analyses and physical interpretations of admitted solutions to quadratic gravity. Here, we demonstrate the utility of such an approach by proving general propositions restricting the spacetime geometry under assumptions on a specific algebraic type of curvature tensors.

AdS–dS stationary rotating black hole exact solution within Einstein-nonlinear electrodynamics

In this report the exact rotating charged black hole solution to the Einstein–nonlinear electrodynamics theory with a cosmological constant is presented. This black hole is equipped with mass, rotation parameter, electric and magnetic charges, cosmological constant Λ, and three parameters due to the nonlinear electrodynamics: β is associated to the potential vectors Aμ and ⋆Pμ, and two constants, F0 and G0, due to the presence of the invariants F and G in the Lagrangian L(F(xμ),G(xμ)). This solution is of Petrov type D, characterized by the Weyl tensor eigenvalue Ψ2, the traceless Ricci tensor eigenvalue S=2Φ(11), and the scalar curvature R; it allows for event horizons, exhibits a ring singularity and fulfils the energy conditions. Its Maxwell limit is the de Sitter-Anti–de Sitter–Kerr–Newman black hole solution.

New gap results on n‐dimensional static manifolds

AbstractIn this note, we obtain new classification results for static and V‐static manifolds under pinching conditions of two different kinds. Firstly, we prove a classification result in higher dimensions assuming a condition of nonnegative Ricci curvature. This extends some recently obtained gap results on the traceless Ricci tensor. Furthermore, we employ the construction of an Einstein manifold associated to a static manifold in order to obtain a classification result in higher dimensions under a pinching condition on the boundary area.

Exact solutions of nonlocal gravity in a class of almost universal spacetimes

We study exact solutions of the infinite derivative gravity with null radiation which belong to the class of almost universal Weyl type III/N Kundt spacetimes. This class is defined by the property that all rank-2 tensors ${B_{ab}}$ constructed from the Riemann tensor and its covariant derivatives have traceless part of type N of the form $\mathcal{B}(\square)S_{ab}$ and the trace part constantly proportional to the metric. Here, $\mathcal{B}(\square)$ is an analytic operator and $S_{ab}$ is the traceless Ricci tensor. We show that the convoluted field equations reduce to a single non-local but linear equation, which contains only the Laplace operator $\triangle$ on 2-dimensional spaces of constant curvature. Such a non-local linear equation is always exactly solvable by eigenfunction expansion or using the heat kernel method for the non-local form-factor $\exp(-\ell^2\triangle)$ (with $\ell$ being the length scale of non-locality) as we demonstrate on several examples. We find the non-local analogues of the Aichelburg--Sexl and the Hotta--Tanaka solutions, which describe gravitational waves generated by null sources propagating in Minkowski, de Sitter, and anti-de Sitter spacetimes. They reduce to the solutions of the local theory far from the sources or in the local limit, ${\ell\to0}$. In the limit ${\ell\to\infty}$, they become conformally flat. We also discuss possible hints suggesting that the non-local solutions are regular at the locations of the sources in contrast to the local solutions; all curvature components in the natural null frame are finite and specifically the Weyl components vanish.

Einstein gravity from Conformal Gravity in 6D

We extend Maldacena’s argument, namely, obtaining Einstein gravity from Conformal Gravity, to six dimensional manifolds. The proof relies on a particular combination of conformal (and topological) invariants, which makes manifest the fact that 6D Conformal Gravity admits an Einstein sector. Then, by taking generalized Neumann boundary conditions, the Conformal Gravity action reduces to the renormalized Einstein-AdS action. These restrictions are implied by the vanishing of the traceless Ricci tensor, which is the defining property of any Einstein spacetime. The equivalence between Conformal and Einstein gravity renders trivial the Einstein solutions of 6D Critical Gravity at the bicritical point.

Regularity conditions for spherically symmetric solutions of Einstein-nonlinear electrodynamics equations

In this report, the regularity conditions at the center for static spherically symmetric (SSS) solutions of the Einstein equations coupled to nonlinear electrodynamics (NLE) are established. The NLE is derived from a Lagrangian L=L(F) depending on the electromagnetic invariant F=FμνFμν∕4. The Einstein-NLE field equations for SSS metrics can be analyzed from the point of view of Euler equations; the general electrically charged SSS solution depending linearly on the electric field E=q0Frt is derived. SSS metrics are characterized by the following independent Riemann tensor invariants: the traceless Ricci (TR) tensor eigenvalue S, the Weyl tensor eigenvalue Ψ2 and the scalar curvature R. Regular solutions are characterized by the finite behavior of these curvature invariants in the whole spacetime. NLE SSS solutions are regular at the center r=0 with limr→0{Ψ2,S,R}→{0,0,(0,4Λ+4L(0))}, if and only if the metric function Q(r) and the electric field q0Frt≕E behave as {Q,Q̇,Q̈}→{0,0,2} and {E,Ė,Ë}→{0,0,0}, as r→0, i.e., the electric field and its first and second order derivatives vanish at the center where the metric asymptotically approaches to the flat or conformally flat de Sitter–Anti de Sitter (dS–AdS) spacetimes. Moreover, this family of solutions may exhibit different asymptotic behavior at spatial infinity such as the Reissner–Nordström (Maxwell) asymptotic, or present the dS–AdS or other kind of asymptotic. Pure magnetic NLE SSS solutions shear the single magnetic invariant 2Fm=h02∕r4, thus they are singular in the magnetic field and may exhibit a regular flat or (A)dS behavior at the center.

Derivative couplings in gravitational production in the early universe

Gravitational particle production in the early universe is due to the coupling of matter fields to curvature. This coupling may include derivative terms that modify the kinetic term. The most general first order action contains derivative couplings to the curvature scalar and to the traceless Ricci tensor, which can be dominant in the case of (pseudo-)Nambu-Goldstone bosons or disformal scalars, such as branons. In the presence of these derivative couplings, the density of produced particles for the adiabatic regime in the de Sitter phase (which mimics inflation) is constant in time and decays with the inverse effective mass (which in turn depends on the coupling to the curvature scalar). In the reheating phase following inflation, the presence of derivative couplings to the background curvature modifies in a nontrivial way the gravitational production even in the perturbative regime. We also show that the two couplings — to the curvature scalar and to the traceless Ricci tensor — are drastically different, specially for large masses. In this regime, the production becomes highly sensitive to the former coupling while it becomes independent of the latter.

Rigidity Theorems of Complete Kähler-Einstein Manifolds and Complex Space Forms

We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete Kahler manifolds. We derive some elliptic differential inequalities from Weitzenb¨ock formulas for the traceless Ricci tensor of Kahler manifolds with constant scalar curvature and the Bochner tensor of Kahler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several Lp and L∞ pinching results are established to characterize Kahler-Einstein manifolds among Kahler manifolds with constant scalar curvature and complex space forms among Kahler-Einstein manifolds. Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact Kahler manifolds and noncompact Kahler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge, these kinds of results have not been reported.

Convergence of Riemannian 4-manifolds with L2L^{2}-curvature bounds

Abstract In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing L 2 {L^{2}} -norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1 we consider a sequence of closed Riemannian 4-manifolds, whose L 2 {L^{2}} -norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a uniform diameter bound, we prove that there exists a subsequence that converges with respect to the Gromov–Hausdorff topology to a flat manifold. In Theorem 1.2 we consider a sequence of closed Riemannian 4-manifolds, whose L 2 {L^{2}} -norm of the Riemannian curvature tensor is uniformly bounded from above, and whose L 2 {L^{2}} -norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the Euclidean situation, and a uniform diameter bound, we show that there exists a subsequence which converges in the Gromov–Hausdorff sense to an Einstein manifold. In order to prove Theorem 1.1 and Theorem 1.2, we use a smoothing technique, which is called L 2 {L^{2}} -curvature flow. This method was introduced by Jeffrey Streets. In particular, we use his “tubular averaging technique” in order to prove distance estimates of the L 2 {L^{2}} -curvature flow, which only depend on significant geometric bounds. This is the content of Theorem 1.3.

On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature – Submanifolds versions

In this paper, from the viewpoint of submanifold theory, we study the isolation phenomena of Riemannian manifolds with constant scalar curvature and vanishing Weyl conformal curvature tensor. Firstly, for any locally strongly convex affine hyperspheres in an (n+1)-dimensional equiaffine space Rn+1 with constant scalar curvature, we prove an inequality involving the traceless Ricci tensor, the Pick invariant and the scalar curvature. The inequality is optimal and we can further completely classify the affine hyperspheres which realize the equality case of the inequality. Secondly, and analogously, for Lagrangian minimal submanifolds of the complex projective space CPn equipped with the Fubini–Study metric, under the condition that the Weyl conformal curvature tensor vanishes, we establish a similar but reverse inequality involving the traceless Ricci tensor, the scalar curvature and the squared norm of the second fundamental form. The inequality is also optimal and we can further completely classify the submanifolds which realize the equality case of the inequality.

On the asymptotically Poincaré-Einstein 4-manifolds with harmonic curvature

Abstract In this paper, we discuss the mass aspect tensor and the rigidity of an asymptotically Poincare-Einstein (APE) 4-manifold with harmonic curvature. We prove that the trace-free part of the mass aspect tensor of an APE 4-manifold with harmonic curvature and normalized Einstein conformal infinity is zero. As to the rigidity, we first show that a complete noncompact Riemannian 4-manifold with harmonic curvature and positive Yamabe constant as well as a L 2 -pinching condition is Einstein. As an application, we then obtain that an APE 4-manifold with harmonic curvature and positive Yamabe constant is isometric to the hyperbolic space provided that the L 2 -norm of the traceless Ricci tensor or the Weyl tensor is small enough and the conformal infinity is a standard round 3-sphere.

On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces

4-dimensional spaces equipped with 2-dimensional (complex holomorphic or real smooth) completely integrable distributions are considered. The integral manifolds of such distributions are totally null and totally geodesics 2-dimensional surfaces which are called the null strings. Properties of congruences (foliations) of such 2-surfaces are studied. Some relations between properties of congruences of null strings, Petrov-Penrose types of the SD Weyl spinor, and algebraic types of the traceless Ricci tensor are analyzed.

Exact solutions to quadratic gravity

Since all Einstein spacetimes are vacuum solutions to quadratic gravity in four dimensions, in this paper we study various aspects of non-Einstein vacuum solutions to this theory. Most such known solutions are of traceless Ricci and Petrov type N with a constant Ricci scalar. Thus we assume the Ricci scalar to be constant which leads to a substantial simplification of the field equations. We prove that a vacuum solution to quadratic gravity with traceless Ricci tensor of type N and aligned Weyl tensor of any Petrov type is necessarily a Kundt spacetime. This will considerably simplify the search for new non-Einstein solutions. Similarly, a vacuum solution to quadratic gravity with traceless Ricci type III and aligned Weyl tensor of Petrov type II or more special is again necessarily a Kundt spacetime. Then we study the general role of conformal transformations in constructing vacuum solutions to quadratic gravity. We find that such solutions can be obtained by solving one non-linear partial differential equation for a conformal factor on any Einstein spacetime or, more generally, on any background with vanishing Bach tensor. In particular, we show that all geometries conformal to Kundt are either Kundt or Robinson-Trautman, and we provide some explicit Kundt and Robinson-Trautman solutions to quadratic gravity by solving the above mentioned equation on certain Kundt backgrounds.

Gravitational Field Equations on Fefferman Space–Times

The total space \({\mathfrak M} \approx {\mathbb H}_1 \times S^1\) of the canonical circle bundle over the 3-dimensional Heisenberg group \({\mathbb H}_1\) is a space–time with the Lorentzian metric \(F_{\theta _0}\) (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form \(\theta _0\) on \({\mathbb H}_1\). The matter and energy content of \(\mathfrak M\) is described by the energy-momentum tensor \({T}_{\mu \nu }\) (the trace-less Ricci tensor of \(F_{\theta _0}\)) as an effect of the non flat nature of Feferman’s metric \(F_{\theta _0}\). We study the gravitational field equations \(R_{\mu \nu } - (1/2) \, R \, g_{\mu \nu } = {T}_{\mu \nu }\) on \({\mathfrak M}\). We consider the first order perturbation \(g = F_{\theta _0} + \epsilon \, h\), \(\epsilon<< 1\), and linearize the field equations about \(F_{\theta _0}\). We determine a Lorentzian metric g on \({\mathfrak M}\) which solves the linearized field equations corresponding to a diagonal perturbation h.

Classification of the Traceless Ricci Tensor in 4-dimensional Pseudo-Riemannian Spaces of Neutral Signature

The traceless Ricci tensor $C_{ab}$ in 4-dimensional pseudo-Riemannian spaces equipped with the metric of the neutral signature is analyzed. Its algebraic classification is given. This classification uses the properties of $C_{ab}$ treated as a matrix. The Petrov-Penrose types of Pleba\'nski spinors associated with the traceless Ricci tensor are given. Finally, the classification is compared with a similar classification in the complex case.

Congruences of Null Strings and Their Relations with Weyl Tensor and Traceless Ricci Tensor

Congruences of Null Strings and Their Relations with Weyl Tensor and Traceless Ricci Tensor

Kundt solutions of minimal massive 3D gravity

We construct Kundt solutions of Minimal Massive Gravity theory and show that, similar to Topologically Massive Gravity (TMG), most of them are constant scalar invariant (CSI) spacetimes, that correspond to deformations of round and warped (A)dS. We also find an explicit non-CSI Kundt solution at the merger point. Finally, we give their algebraic classification with respect to the traceless Ricci tensor (Segre classification) and show that their Segre-types match with the types of their counterparts in TMG.

On the structure of conformally flat Riemannian manifolds

In this paper we will prove vanishing and finiteness theorems for L2-harmonic 1-forms on a locally conformally flat Riemannian manifold which satisfies an integral pinching condition on the traceless Ricci tensor, and for which the scalar curvature is non-positive or satisfies some integral pinching conditions. Based on these vanishing and finiteness theorems, combining with the work of Li–Tam, we can obtain some one-end and finite ends theorems.

AdS-plane wave andpp-wave solutions of generic gravity theories

We construct the AdS-plane wave solutions of generic gravity theory built on the arbitrary powers of the Riemann tensor and its derivatives in analogy with the pp-wave solutions. In constructing the wave solutions of the generic theory, we show that the most general two tensor built from the Riemann tensor and its derivatives can be written in terms of the traceless-Ricci tensor. Quadratic gravity theory plays a major role; therefore, we revisit the wave solutions in this theory. As examples to our general formalism, we work out the six-dimensional conformal gravity and its nonconformal deformation as well as the tricritical gravity, the Lanczos-Lovelock theory, and string-generated cubic curvature theory.