Ricci tensor

In this paper, we study Ricci-flat and Einstein Lorentzian multiply warped products. We also consider the case of having constant scalar curvatures for this class of warped products. Finally, after we introduce a new class of spacetimes called as generalized Kasner space-times, we apply our results to this kind of space-times as well as other relativistic space-times, i.e., Reissner-Nordström, Kasner space-times, Bañados-Teitelboim-Zanelli and de Sitter Black hole solutions.

A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity by Lovelock.

I discuss Einstein's path-breaking November 1915 General Relativity papers. I show that Einstein's field equations of November 25, 1915 with an additional term on the right hand side involving the trace of the energy-momentum tensor appear to have sprung from his first November 1915 paper: the November 4, 1915 equations. Second paper among three papers.

We study Riemannian manifolds, subject to a prescribed symmetry inheritance, defined by L~g~ = 20tg2, where g2, c~, and L~ are geometric/physical object, function, and Lie derivative operator with respect to a vector field ~. In this paper, we set ~ = Riemann curvature tensor or Ricci tensor and obtain several new results relevant to physically significant material curves, proper eonformal and proper noneonformal symmetries. In particular, we concentrate on a time-like Rieci inheritance vector parallel to the velocity vector of a perfect fluid spacetime. We claim new and physically relevant equations of state. All key results are supported by physical examples, including the Friedman-Robertson-Walker universe models. In general, this paper opens a new area of research on symmetry inheritance with a potential for further applications in mathematical physics. (1991). 53C50, 53C21, 83C15, 83C20.

I discuss Albert Einstein's 1916 General Theory of Relativity. I show that in Einstein's 1916 review paper, "the Foundation of the General Theory of Relativity", he derived his November 25, 1915 field equations with an additional term on the right hand side involving the trace of the energy-momentum tensor (he posed the condition square root -g=1) using the equations he presented on November 4, 1915. Series of papers: Final paper.

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching generalization to special connections. A

We introduce the concept of induced scalar curvature of a class C[M ] of lightlike hypersurfaces (M, g, S(T M )), of a Lorentzian manifold, such that M admits a canonical screen distribution S(T M ), a canonical lightlike transversal vector bundle and an induced symmetric Ricci tensor. We prove that there exists such a class C[M ] of a globally hyperbolic warped product spacetime [3] of general relativity. In particular, we calculate scalar curvature of a member of C[M ] in a globally hyperbolic spacetime of constant curvature, supported by an example. MSC: 53C20; 53C50; 53B50 Subj. Class: Differential Geometry; General relativity

We review the status of the fourth-order quartic in the spacetime curvature terms induced by superstrings/M-theory compactified on a warped torus in the leading order with respect to the Regge slope parameter, and study their nonperturbative impact on the evolution of the Hubble scale in the context of the four-dimensional FRW cosmology. After taking into account the quantum ambiguities in the definition of the off-shell superstring effective action, we propose the generalized Friedmann equations, find the existence of their de Sitter exact inflationary solutions without a spacetime singularity, and constrain the ambiguities by demanding stability and the scale factor duality invariance of our solutions. The most naive Bel-Robinson tensor squared quartic terms are ruled out, thus giving the evidence for the necessity of extra quartic Ricci tensordependent terms in the off-shell gravitational effective action for superstrings. Our methods are generalizable to the higher orders in the spacetime curvature.

We study a natural generalization of the concepts of torsion and Ricci tensor for a nonlinear connection on a fibred manifolds, with respect to a given fibred soldering form. Our results are achieved by means of the differentials and codifferentials induced by the Frölicher-Nijenhuis graded Lie algebra of tangent valued forms.

Before developing his 1915 General Theory of Relativity, Einstein held the "Entwurf" theory. Tullio Levi-Civita from Padua, one of the founders of tensor calculus, objected to a major problematic element in this theory, which reflected its global problem: its field equations were restricted to an adapted coordinate system. Einstein proved that his gravitational tensor was a covariant tensor for adapted coordinate systems. In an exchange of letters and postcards that began in March 1915 and ended in May 1915, Levi-Civita presented his objections to Einstein's above proof. Einstein tried to find ways to save his proof, and found it hard to give it up. Finally Levi-Civita convinced Einstein about a fault in his arguments. However, only in spring 1916, long after Einstein had abandoned the 1914 theory, did he finally understand the main problem with his 1914 gravitational tensor. In autumn 1915 the G\"ottingen brilliant mathematician David Hilbert found the central flaw in Einstein's 1914 derivation. On March 30, 1916, Einstein sent to Hilbert a letter admitting, "The error you found in my paper of 1914 has now become completely clear to me".

On compact balanced Hermitian manifolds we obtain obstructions to the existence of harmonic 1-forms, 9-harmonic (1,0)-forms and holomorphic (1,0)-forms in terms of the Ricci tensors with respect to the Riemannian curvature and the Hermitian curvature. Vanishing of the first Dolbeault cohomology groups of the twistor space of a compact irreducible hyper Kahler manifold is shown. A necessary and sufficient condition the (1,0)-part of a harmonic 1-form to be holomorphic and vice versa, a real 1-form with a holomorphic (1,0)-part to be harmonic are found.

The exact static spherically symmetric solutions for pure-aether theory and Einsteinaether theory are presented. It is shown that both theories can deliver the Schwarzschild metric, but only the Einstein-aether theory contains solutions with "almost-Schwarzschild" metrics that satisfy Einstein's experiments. Two specific solutions are of special interest: one in pure-aether theory that derives the attractive nature of gravitation as a result of Minskowski signature of the metric, and one -the Jacobson solution-of Einstein-aether theory with "almost-Schwarzschild" metric and non-zero Ricci tensor.

In this paper we obtain generalized Keller-Osserman conditions for wide classes of differential inequalities on weighted Riemannian manifolds of the form $L u\geq b(x) f(u) \ell(|\nabla u|)$ and $L u\geq b(x) f(u) \ell(|\nabla u|) - g(u) h(|\nabla u|)$, where $L$ is a non-linear diffusion-type operator. Prototypical examples of these operators are the $p$-Laplacian and the mean curvature operator. While we concentrate on non-existence results, in many instances the conditions we describe are in fact necessary for non-existence. The geometry of the underlying manifold does not affect the form of the Keller-Osserman conditions, but is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions $b$ and $\ell$. We also describe a weak maximum principle related to inequalities of the above form which extends and improves previous results valid for the $\vp$-Laplacian.

Gravitational instantons, solutions to the euclidean Einstein equations, with topology $R^3 XS^1$ arise naturally in any discussion of finite temperature quantum gravity. This Letter shows that all such instantons (irrespective of their interior behaviour) must have the same asymptotic structure as the Schwarzschild instanton. Using this, it can be shown that if the Ricci tensor of the manifold is non-negative it must be flat. One special case is when the Ricci tensor vanishes; hence one can conclude that there is no nontrivial vacuum gravitational instanton. This result has uses both in quantum and classical gravity. It places a significant restriction on the instabilities of hot flat space. It also can be used to show that any static vacuum Lorentzian Kaluza-Klein solution is flat.

This paper is dedicated to the statistical analysis of the space of multivariate normal distributions with an application to the processing of Diffusion Tensor Images (DTI). It relies on the differential geometrical properties of the underlying parameters space, endowed with a Riemannian metric, as well as on recent works that led to the generalization of the normal law on Riemannian manifolds. We review the geometrical properties of the space of multivariate normal distributions with zero mean vector and focus on an original characterization of the mean, covariance matrix and generalized normal law on that manifold. We extensively address the derivation of accurate and efficient numerical schemes to estimate these statistical parameters. A major application of the present work is related to the analysis and processing of DTI datasets and we show promising results on synthetic and real examples.

The coupling of the Higgs field through the Ricci tensor, put forward by Balakrishna and Wali, is derived using a conformal rescaling of the metric. Earlier results on "Bogomolny-type" equations in curved space, by Comtet, and others, are recovered. The procedure can be generalized to any static background metric.

We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds (B, gB) and (F, gF ) furnished with metrics of the form c 2 gB ⊕ w 2 gF and, in particular, of the type w 2µ gB ⊕ w 2 gF , where c, w : B → (0, ∞) are smooth functions and µ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If (B, gB) is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave-convex nonlinearities and singular partial differential equations of the Lichnerowicz-York type among others.

Geometrical characterizations are given for the tensor R · S, where S is the Ricci tensor of a (semi-)Riemannian manifold (M, g) and R denotes the curvature operator acting on S as a derivation, and of the Ricci Tachibana tensor ∧ g ·S, where the natural metrical operator ∧ g also acts as a derivation on S. As a combination, the Ricci curvatures associated with directions on M, of which the isotropy determines that M is Einstein, are extended to the Ricci curvatures of Deszcz associated with directions and planes on M, and of which the isotropy determines that M is Ricci pseudo-symmetric in the sense of Deszcz.

We use the framework used by Bakry and Emery in their work on logarithmic
Sobolev inequalities to define a notion of coarse Ricci curvature on smooth
metric measure spaces alternative to the notion proposed by Y. Ollivier.
\ This function can be used to recover the Ricci tensor on smooth Riemannian
manifolds by the formula

We analyze black hole thermodynamics in a generalized theory of gravity whose Lagrangian is an arbitrary function of the metric, the Ricci tensor and a scalar field. We can convert the theory into the Einstein frame via a "Legendre" transformation or a conformal transformation. We calculate thermodynamical variables both in the original frame and in the Einstein frame, following the Iyer-Wald definition which satisfies the first law of thermodynamics. We show that all thermodynamical variables defined in the original frame are the same as those in the Einstein frame, if the spacetimes in both frames are asymptotically flat, regular and possess event horizons with non-zero temperatures. This result may be useful to study whether the second law is still valid in the generalized theory of gravity.

We introduce the concept of a base conformal warped product of two pseudo-Riemannian manifolds. We also define a subclass of this structure called as a special base conformal warped product. After, we explicitly mention many of the relevant fields where metrics of these forms and also considerations about their curvature related properties play important rolls. Among others, we cite general relativity, extra-dimension, string and super-gravity theories as physical subjects and also the study of the spectrum of Laplace-Beltrami operators on p-forms in global analysis. Then, we give expressions for the Ricci tensor and scalar curvature of a base conformal warped product in terms of Ricci tensors and scalar curvatures of its base and fiber, respectively. Furthermore, we introduce specific identities verified by particular families of, either scalar or tensorial, nonlinear differential operators on pseudo-Riemannian manifolds. The latter allow us to obtain new interesting expressions for the Ricci tensor and scalar curvature of a special base conformal warped product and it turns out that not only the expressions but also the analytical approach used are interesting from the physical, geometrical and analytical point of view. Finally, we analyze, investigate and characterize possible solutions for the conformal and warping factors of a special base conformal warped product, which guarantee that the corresponding product is Einstein. Besides, we apply these results to a generalization of the Schwarzschild metric.

We introduce a new kind of Riemannian manifold that includes weakly-, pseudo-and pseudo projective-Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named weakly Z symmetric and denoted by (W ZS)n. If the Z tensor is singular we give conditions for the existence of a proper concircular vector. For non singular Z tensor, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally harmonic case, and the general form of the Ricci tensor. For conformally flat (W ZS)n manifolds, we derive the local form of the metric tensor. DateHere we define the Ricci tensor as R kl = −R mkl m and the scalar curvature as R = g ij R ij .

Killing vector fields in three dimensions play an important role in the construction of the related spacetime geometry. In this work we show that when a three-dimensional geometry admits a Killing vector field then the Ricci tensor of the geometry is determined in terms of the Killing vector field and its scalars. In this way we can generate all products and covariant derivatives at any order of the Ricci tensor. Using this property we give ways to solve the field equations of topologically massive gravity (TMG) and new massive gravity (NMG) introduced recently. In particular when the scalars of the Killing vector field (timelike, spacelike and null cases) are constants then all three-dimensional symmetric tensors of the geometry, the Ricci and Einstein tensors, their covariant derivatives at all orders, and their products of all orders are completely determined by the Killing vector field and the metric. Hence, the corresponding three-dimensional metrics are strong candidates for solving all higher derivative gravitational field equations in three dimensions.

We prove that some Riemannian manifolds with boundary satisfying an explicit integral pinching condition are spherical space forms. More precisely, we show that three-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the L 2 -norm of the scalar curvature and the L 2 -norm of the Ricci tensor are spherical space forms with totally geodesic boundary. Moreover, we prove also that four-dimensional Riemannian manifolds with umbilic boundary, positive Yamabe invariant and an explicit integral pinching between the total integral of the (Q, T )-curvature and the L 2 -norm of the Weyl curvature are spherical space forms with totally geodesic boundary. As a consequence, we show that a certain conformally invariant operator, which plays an important role in Conformal Geometry, is non-negative and has trivial kernel if the Yamabe invariant is positive and verifies a pinching condition together with the total integral of the (Q, T )-curvature. As an application of the latter spectral analysis, we show the existence of conformal metrics with constant Q-curvature, constant T -curvature, and zero mean curvature under the latter assumptions.

In this paper we discuss new measures for connectivity analysis of brain white matter, using MR diffusion tensor imaging. Our approach is based on Riemannian geometry, the viability of which has been demonstrated by various researchers in foregoing work. In the Riemannian framework bundles of axons are represented by geodesics on the manifold. Here we do not discuss methods to compute these geodesics, nor do we rely on the availability of geodesics. Instead we propose local measures which are directly computable from the local DTI data, and which enable us to preselect viable or exclude uninteresting seed points for the potentially time consuming extraction of geodesics. If geodesics are available, our measures can be readily applied to these as well.

Abstract: In this work we discuss the exact solution to the algebraic equation associated to the Ricci tensor in the quadratic f(R,Q) extension of Palatini gravity. We show that an exact solution always exists, and in the general case it can be found by matrix diagonalization. Furthermore, the general implications of the solution are analysed in detail, including the generation of an effective cosmological constant, and the recovery of the f(R) and f(Q) theories as particular cases in the corresponding limit. In addition, it is proposed a power series expansion of the solution which is successfully applied to the case of the electromagnetic field. We show that this power series expansion may be useful to deal perturbatively with some problems in the context of Palatini gravity.

... Marco Ferrarist, Mauro Francaviglial and Guido MagnanoP t Dipartimento di Matematica, Universita di Cagliari, via Ospedale 72,09100 Cagliari, Italy $: lstituto di Fisica ... In a very recent paper [9], Cecotti has shown how to generalise Whitt's result to the case of supergravity. ...

A conceptual summary is given of a deterministic unified field and particle theory (the metron model) developed in more mathematical detail in a four-part paper published in Physics Essays (1996/97). The model is developed from Einsteins vacuum gravitational equations, Ricci tensor $R_{LM}=0$, in a higher dimensional space. It is postulated that the equations support soliton-type solutions (metrons) which reproduce all the basic field equations of quantum field theory, including not only the Maxwell-Dirac-Einstein system, but also all fields and symmetries of the Standard Model. Bell's theorem on the non-existence of hidden-variable models of quantum phenomena is circumvented through the time-reversal symmetry of all interactions on microphysical scales. The model, when completed, should yield all particle properties and universal physical constants from first principles.

We describe all almost contact metric, almost hermitian and $G_2$-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the $\nabla$-parallel spinors. In particular, we obtain solutions of the type II string in dimension $n=5,6$ and 7.

We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with dd c -harmonic Kähler form and positive (1, 1)-part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such that the Ricci tensor of the canonical Weyl structure is positive. As a corollary we obtain that any such surface must be rational with c 2 1 > 0. As an application, the pth Dolbeault cohomology groups of a left-invariant complex structure compatible with a bi-invariant metric on a compact even dimensional Lie group are computed. Running title: Vanishing theorems on Hermitian manifolds Theorem 1.1 Let (M, g, J) be a compact 2n-dimensional (n > 1) Hermitian manifold with Kähler form Ω. Suppose that Ω is dd c -harmonic, i.e. dd c Ω = 0 and (dd c ) * Ω = 0. Suppose also that the (1, 1)-part of the Ricci form of the Bismut connection is non-negative everywhere on M . a) Then every ∂-harmonic (0,p)-form, p = 1, 2, ..., n, must be parallel with respect to the Bismut connection. b) If moreover the (1, 1)-part of the Ricci form of the Bismut connection is strictly positive at some point, then the cohomology groups H p (M, O) vanish for p = 1, 2, ..., n.

We shall establish in the context of adapted differential geometry on the path space P mo (M) a Weitzenböck formula which generalizes that in (A. B. Cruzeiro and P. Malliavin, J. Funct. Anal. 177 (2000), 219-253), without hypothesis on the Ricci tensor. The renormalized Ricci tensor will be vanished. The connection introduced in (A. B.

The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a group Gr of isometries acting on s-dimensional orbits are given. This provides the list of (abstract) groups that can act isometrically and maximally on such metrics. The conditions are expressed in terms of the eigenvalues and eigenvectors of the Ricci tensor. In any case, the order of differentiability of these data necessary to determine the isometry group is less than 4.