Perturbative modes and black hole entropy in f(Ricci) gravity

f(Ricci) gravity is a special kind of higher curvature gravity whose bulk Lagrangian density is the trace of a matrix-valued function of the Ricci tensor. It is shown that under some mild constraints, f(Ricci) gravity admits Einstein manifolds as exact vacuum solutions, and can be ghost-free and tachyon-free around maximally symmetric Einstein vacua. It is also shown that the entropy for spherically symmetric black holes in f(Ricci) gravity calculated via the Wald method and the boundary Noether charge approach are in good agreement.     

A Complete Classification of Curvature Collineations of Cylindrically Symmetric Static Metrics

Curvature collineations are symmetry directions for the Riemann tensor, as isometries are for the metric tensor and Ricci collineations are for the Ricci tensor. Complete listings of many metrics possessing some minimal symmetry have been given for a number of symmetry groups for the latter two symmetries. It is shown that a claimed complete listing of cylindrically symmetric static metrics by their curvature collineations [1] was actually incomplete and is completed here. It turns out that in this complete list, unlike the previous claim, there are curvature collineations that are distinct from the set of isometries and of Ricci collineations. The physical interpretation of some of the metrics obtained is given.     

Ricci Curvature-Based Semi-Supervised Learning on an Attributed Network

In recent years, on the basis of drawing lessons from traditional neural network models, people have been paying more and more attention to the design of neural network architectures for processing graph structure data, which are called graph neural networks (GNN). GCN, namely, graph convolution networks, are neural network models in GNN. GCN extends the convolution operation from traditional data (such as images) to graph data, and it is essentially a feature extractor, which aggregates the features of neighborhood nodes into those of target nodes. In the process of aggregating features, GCN uses the Laplacian matrix to assign different importance to the nodes in the neighborhood of the target nodes. Since graph-structured data are inherently non-Euclidean, we seek to use a non-Euclidean mathematical tool, namely, Riemannian geometry, to analyze graphs (networks). In this paper, we present a novel model for semi-supervised learning called the Ricci curvature-based graph convolutional neural network, i.e., RCGCN. The aggregation pattern of RCGCN is inspired by that of GCN. We regard the network as a discrete manifold, and then use Ricci curvature to assign different importance to the nodes within the neighborhood of the target nodes. Ricci curvature is related to the optimal transport distance, which can well reflect the geometric structure of the underlying space of the network. The node importance given by Ricci curvature can better reflect the relationships between the target node and the nodes in the neighborhood. The proposed model scales linearly with the number of edges in the network. Experiments demonstrated that RCGCN achieves a significant performance gain over baseline methods on benchmark datasets.     

Locally Homogeneous Four-Dimensional Manifolds of Signature (2,2)

In this paper we consider 4-dimensional neutral-signature curvature models and obtain the complete classification of the Ricci operator. We then consider the property of curvature homogeneity for the above manifolds and prove that every complete, connected and simply connected 1-curvature homogeneous 4-dimensional manifold of signature (2,2) with a non-degenerate Ricci operator is isometric to a four-dimensional Lie group equipped with a left invariant neutral metric. We also classify Ricci-parallel curvature homogeneous 4-dimensional manifolds of signature (2,2).     

Mean Curvature Flow in a Ricci Flow Background

Following work of Ecker (Comm Anal Geom 15:1025–1061, 2007), we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman’s modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton’s differential Harnack expression for the mean curvature flow in Euclidean space. We also derive the evolution equations for the second fundamental form and the mean curvature, under a mean curvature flow in a Ricci flow background. In the case of a gradient Ricci soliton background, we discuss mean curvature solitons and Huisken monotonicity.     

TheB (3) field as a link between gravitation and electromagnetism in the vacuum

The emergence of theB ⁽³⁾ field in vacuo has shown that electromagnetism is non-Abelian and similar in structure to gravitation. In this paper the Christoffel symbol used in general relativity is developed for electromagnetism in curvilinear coordinates: The former becomes describable as the antisymmetric part of the gravitational Ricci tensor. Therefore gravitation and electromagnetism are respectively the symmetric and antisymmetric parts of thesame Ricci tensor within a proportionality factor. Both fields are obtained from the Riemann curvature tensor, both are expressions of curvature in spacetime.     

Letter: Comment on Ricci Collineations for Spherically Symmetric Space-Times

It is shown that the results of the paper Contreras, G., Nunez, L. A., Percoco, U. Ricci Collineations for Non-degenerate, Diagonal and Spherically Symmetric Ricci Tensors (2000). Gen. Rel. Grav. 32, 285-294 concerning the Ricci Collineations in spherically symmetric space-times with non-degenerate and diagonal Ricci tensor do not cover all possible cases. Furthermore the complete algebra of Ricci Collineations of certain Robertson-Walker metrics of vanishing spatial curvature are given.     

On Ricci tensors of Randers metrics

In this paper, we study Randers metrics and find a condition on the Ricci tensors of these metrics for being Berwaldian. This generalizes Shen’s Theorem which says that every R-flat complete Randers metric is locally Minkowskian. Then we find a necessary and sufficient condition on the Ricci tensors under which a Randers metric of scalar flag curvature is of zero flag curvature.     

Constant gravitational fields and redshift of light

The solutions of Einsteins's equations in a constant energy-momentum tensor field are Ricci curvature homogeneous. Convenient perturbations of a Lorentz solvmanifold yield such curvature homogeneous metrics, prescribing redshift of light and singularities.     

Symmetry mappings in Einstein-Maxwell space-times

General properties of Einstein-Maxwell spaces, with both null and nonnull source-free Maxwell fields, are examined when these space-times admit various kinds of symmetry mappings. These include Killing, homothetic and conformal vector fields, curvature and Ricci collineations, and mappings belonging to the family of contracted Ricci collineations. In particular, the behavior of the electromagnetic field tensor is examined under these symmetry mappings. Examples are given of such space-times which admit proper curvature and proper Ricci collineations. Examples are also given of such space-times in which the metric tensor admits homothetic and other motions, but in which the corresponding Lie derivatives of the electromagnetic Maxwell tensor are not just proportional to the Maxwell tensor.On leave from Mathematics Department, Monash University, Clayton, Victoria, 3168, Australia.     

PP-waves with torsion: a metric-affine model for the massless neutrino

In this paper we deal with quadratic metric-affine gravity, which we briefly introduce, explain and give historical and physical reasons for using this particular theory of gravity. We then introduce a generalisation of well known spacetimes, namely pp-waves. A classical pp-wave is a 4-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. This definition was generalised in our previous work to metric compatible spacetimes with torsion and used to construct new explicit vacuum solutions of quadratic metric-affine gravity, namely generalised pp-waves of parallel Ricci curvature. The physical interpretation of these solutions we propose in this article is that they represent a conformally invariant metric-affine model for a massless elementary particle. We give a comparison with the classical model describing the interaction of gravitational and massless neutrino fields, namely Einstein–Weyl theory and construct pp-wave type solutions of this theory. We point out that generalised pp-waves of parallel Ricci curvature are very similar to pp-wave type solutions of the Einstein–Weyl model and therefore propose that our generalised pp-waves of parallel Ricci curvature represent a metric-affine model for the massless neutrino.     

Dust Static Cylindrically Symmetric Solutions in f(R) Gravity

This paper is devoted to investigate non-vacuum solutions of cylindrically symmetric spacetime in the context of metric f(R) gravity. We take dust matter to find energy density of the universe. In particular, we find two exact solutions, which correspond to two f(R) models in each case. The first solution provides constant curvature while the second solution corresponds to non-constant curvature. The functions of the Ricci scalar and energy densities are evaluated in each case.     

Dust Static Cylindrically Symmetric Solutions in f(R) Gravity

This paper is devoted to investigate non-vacuum solutions of cylindrically symmetric spacetime in the context of metric f(R) gravity. We take dust matter to find energy density of the universe. In particular, we find two exact solutions, which correspond to two f(R) models in each case. The first solution provides constant curvature while the second solution corresponds to non-constant curvature. The functions of the Ricci scalar and energy densities are evaluated in each case.     

Riemannian curvature and the classification of the Riemann and Ricci tensors in space-time

Some theorems proved by Thorpe concerning the connection between the critical point structure of the Riemannian (sectional) curvature function and the Petrov classification are extended. A similar function is defined whose critical point structure is connected with the algebraic classification of the Ricci tensor.     

Weyl,curvature, ricci,and metric tensor symmetries

Weyl symmetries for some specific spherically symmetric static spacetimes are derived and compared with metric, Ricci, and curvature tensor symmetries.     

Volume comparison between spacelike hypersurfaces in a Lorentzian manifold with integral Ricci curvature bounds

We obtain the volume comparison between spacelike hypersurfaces in a Lorentzian manifold with integral Ricci and mean curvature bounds. Also we extend volume comparisons to weighted volume comparisons with integral norms of the generalized Ricci tensor.     

On curvature homogeneous three-dimensional Lorentzian manifolds

The aim of this paper is the study of three-dimensional Lorentzian manifolds whose Ricci tensor has three equal constant eigenvalues, whose associated eigenspace is two-dimensional. A complete local classification of this class of curvature homogeneous manifolds is presented. It turns out that, if the eigenvalue is zero, these are exactly the curvature homogeneous manifolds modelled on an indecomposable, non-irreducible Lorentzian symmetric space, which were first studied in Cahen etaal. (1990), and the techniques presented in this paper can therefore be applied to obtain a complete (local) classification of these manifolds, and to construct a number of new examples of such manifolds.     

Concircular Curvature Tensor and Fluid Spacetimes

In the differential geometry of certain F-structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.     

Comparison theorems in Lorentzian geometry and applications to spacelike hypersurfaces

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic operators.     

Relative Lorentzian volume comparison with integral Ricci and scalar curvature bound

A relative Lorentzian volume comparison estimate between spacelike hypersurfaces is studied with the integral curvature bound in terms of Ricci and Scalar curvature which generalize the Bishop–Gromov volume comparison theorem.