Ricci Curvature Research Articles

An axially symmetric spacetime with causality violation in Ricci-inverse gravity

In this paper, Ricci-inverse gravity is investigated. It is an alternative theory of gravity that introduces into the Einstein–Hilbert action an anti-curvature scalar that is obtained from the anti-curvature tensor which is the inverse of the Ricci tensor. An axially symmetric spacetime with causality violation is studied. Two classes of the model are discussed. Different sources of matter are considered. Then a direct relation between the content of matter and causality violation is shown. Our results confirm that Ricci-inverse gravity allows the existence of Closed Time-like Curves (CTCs) that lead to the violation of causality. Furthermore, a comparison is made between the results of general relativity and Ricci-inverse gravity. Other spacetimes, such as Gödel and Gödel-type universes, which are exact solutions of general relativity and allow for causality violations, are also explored in Ricci-inverse gravity framework.

Intermediate Ricci Curvatures and Gromov’s Betti number bound

We consider intermediate Ricci curvatures Ric_k on a closed Riemannian manifold M^n. These interpolate between the Ricci curvature when k=n-1 and the sectional curvature when k=1. By establishing a surgery result for Riemannian metrics with Ric_k>0, we show that Gromov’s upper Betti number bound for sectional curvature bounded below fails to hold for Ric_k>0 when lfloor n/2 rfloor +2 le k le n-1. This was previously known only in the case of positive Ricci curvature (Sha and Yang in J Differ Geom 29(1):95–103, 1989, J Differ Geom 33:127–138, 1991).

Heat kernel estimate for the Laplace-Beltrami operator under Bakry-Émery Ricci curvature condition and applications

We establish a Gaussian upper bound of the heat kernel for the Laplace-Beltrami operator on complete Riemannian manifolds with Bakry-Émery Ricci curvature bounded below. As applications, we first prove an L1-Liouville property for non-negative subharmonic functions when the potential function of the Bakry-Émery Ricci curvature tensor is of at most quadratic growth. Then we derive lower bounds of the eigenvalues of the Laplace-Beltrami operator on closed manifolds. An upper bound of the bottom spectrum is also obtained.

On the mathematical structure of Einstein field equations and the existence of dark fields

In this work, we examine the possible existence of dark fields by showing that an energy‐momentum tensor associated with a dark field can be introduced into Einstein field equations of general relativity, in which the trace of the energy‐momentum of the dark field is identified with the cosmological constant. The introduction of dark fields into Einstein field equations is made possible by establishing field equations for the Ricci curvature tensor, which have similar mathematical structure to Maxwell field equations of electromagnetism. We also establish a system of field equations for the Riemann curvature tensor and investigate the possibility to represent physical systems consisting of dark fields and observable fields as spaces of constant scalar curvature, which are maximally symmetric spaces that admit the maximal number of Killing vectors. As an illustration, we show that if a dark field is considered as a dark fluid, then the pressure associated with the dark field can take negative values if the cosmological constant is assigned with positive values.

Holonomic and Non-Holonomic Geometric Models Associated to the Gibbs–Helmholtz Equation

By replacing the internal energy with the free energy, as coordinates in a “space of observables”, we slightly modify (the known three) non-holonomic geometrizations from Udriste’s et al. work. The coefficients of the curvature tensor field, of the Ricci tensor field, and of the scalar curvature function still remain rational functions. In addition, we define and study a new holonomic Riemannian geometric model associated, in a canonical way, to the Gibbs–Helmholtz equation from Classical Thermodynamics. Using a specific coordinate system, we define a parameterized hypersurface in R4 as the “graph” of the entropy function. The main geometric invariants of this hypersurface are determined and some of their properties are derived. Using this geometrization, we characterize the equivalence between the Gibbs–Helmholtz entropy and the Boltzmann–Gibbs–Shannon, Tsallis, and Kaniadakis entropies, respectively, by means of three stochastic integral equations. We prove that some specific (infinite) families of normal probability distributions are solutions for these equations. This particular case offers a glimpse of the more general “equivalence problem” between classical entropy and statistical entropy.

On Hopf hypersurfaces of [formula omitted] and [formula omitted

In this paper, we study Hopf hypersurfaces of the two Kähler surfaces S2×S2 and H2×H2. First, we classify real hypersurfaces of S2×S2 and H2×H2 with their shape operator A and induced almost contact structure ϕ satisfying Aϕ+ϕA=ρϕ for a constant ρ. Then, we classify real hypersurfaces of H2×H2 with isometric Reeb flow. Finally, we classify Hopf hypersurfaces of S2×S2 and H2×H2 with constant Reeb function as well as with recurrent Ricci tensor.

Para-Kahler and Para-Hermitian Structures on Six-Dimensional Unsolvable Lie Algebras

In this paper, we investigate into the matter of the existence of para-Kahlerian and para-Hermitian structures on six-dimensional unsolvable Lie algebras that are semidirect products. According to the classification results, there are four Lie algebras that are semidirect products of the Lie algebras so(3), sl(2, R) and three soluble Lie algebras A3.1=R3, A3.3 and A3.5. We show that only g=A3.5⋉sl(2, R) has a symplectic structure, and it admits a para-Kahlerian structure of zero Ricci curvature. The paper presents calculated curvature characteristics and the method to find other para-Kahler structures based on deformations of some initial para-Kahler structure. Other Lie algebras admit para-Hermitian structures, i.e. integrable paracomplex structures consistent with the natural non-degenerate 2-form. It follows from the results of the paper that the sixdimensional symplectic Lie algebra must be solvable except for one case when g=A3.5⋉sl(2, R). It complements the well-known result of Chu Bon-Yao that a four-dimensional symplectic Lie algebra must be solvable.

A geometric approach to the modified Milnor problem

The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. We show that a positive answer to the Milnor Problem (modified) is equivalent to the Nilpotency conjecture in Riemannian geometry: given [Formula: see text], there exists a constant [Formula: see text] such that if a compact Riemannian [Formula: see text]-manifold [Formula: see text] satisfies that Ricci curvature [Formula: see text], diameter [Formula: see text] and volume entropy [Formula: see text], then the fundamental group [Formula: see text] is virtually nilpotent. We will verify the Nilpotency conjecture in some cases, and we will verify the vanishing gap phenomena for more cases i.e. if [Formula: see text], then [Formula: see text].

Comprehensive analysis of relativistic embedded class-I exponential compact spheres in f(R, ϕ) gravity via Karmarkar condition

In this article, we use the prominent Karmarkar condition to investigate some novel features of astronomical objects in the f(R, ϕ) gravity; R and ϕ represent the Ricci curvature and the scalar field, respectively. It is worth noting that we classify the exclusive set of modified field equations using the exponential type model of the f(R, ϕ) theory of gravity f(R, ϕ) = ϕ(R + α(e β R − 1)). We show the embedded class-I approach via a static, spherically symmetric spacetime with an anisotropic distribution. To accomplish our objective, we use a particular interpretation of metric potential (g rr ) that has already been given in the literature and then presume the Karmarkar condition to derive the second metric potential. We employ distinct compact stars to determine the values of unknown parameters emerging in metric potentials. To ensure the viability and consistency of our exponential model, we execute distinct physical evolutions, i.e. the graphical structure of energy density and pressure evolution, mass function, adiabatic index, stability, equilibrium, and energy conditions. Our investigation reveals that the observed anisotropic findings are physically appropriate and have the highest level of precision.

Optimal transport and variational Bayesian inference

We apply Lott–Sturm–Villani notion of Ricci curvature as well as McCann–Brenier theorem on optimal transport investigating mean-field variational inference (MFVI). In the continuous case where the probability rules are defined on Riemannian manifolds we obtain sufficient conditions for convexity of the corresponding variational problem. It can be shown that in the presence of symmetry the convexity can provide the preservation of mode. We then consider factorization into a mixed continuous-discrete case. By rescaling the Riemannian structure of the continuous component we have derived sufficient conditions under which the convexity of Kullback-Leibler functional is established. We apply the latter theorem to Bayesian Gaussian Mixture Model (GMM) and we introduce a new class of these models which have the advantage of being convex. It is shown that the MAP for the corrected GMM remains a consistent estimator. Additionally thanks to convexity we are able to prove the mode of MFVI for the new model in asymptotic regimes converges to the right solution. We do not know any other rigorous result for MFVI in statistical learning.

Complete conformal metrics of negative Ricci curvature in open manifolds

In this paper, we study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor in open manifolds. We prove the existence of such metrics by solving a class of fully nonlinear elliptic equations with the infinity Dirichlet boundary condition.

Cure-GNN: A Robust Curvature-Enhanced Graph Neural Network Against Adversarial Attacks

Graph neural networks (GNNs) are a specialized type of deep learning models on graphs by learning aggregations over neighbor nodes. However, recent studies reveal that the performance of GNNs are severely deteriorated by injecting adversarial examples. Hence, improving the robustness of GNNs is of significant importance. Prior works are devoted to reducing the influence of direct adversaries which are adversarial attacks by positioning a node's one-hop neighbors, yet these approaches are limited in protecting GNNs from indirect adversarial attacks within a node's multi-hop neighbors. In this work, we approach this problem from a new angle by exploring the graph Ricci curvature, which can characterize the relationships of both direct and indirect links from any two nodes' neighborhoods in the Riemannian space. We first investigate the distinguishable properties of adversarial attacks with graph Ricci curvature distribution. Then, a novel defense framework called Cure-GNN is proposed to detect and mitigate adversarial effects. Cure-GNN discerns the distinction between adversarial edges and normal edges via computing curvature, and merges it into the node features reconstructed by a residual learning framework. Extensive experiments over real-world datasets on node classification task demonstrate the efficacy of Cure-GNN and achieves superiority to the state-of-the-arts without incurring high complexity.

Geometric and Physical Properties of Curvature Tensors -A Review

Bernard Riemann was the first to define curvature tensor. Most of the curvature tensors are defined with the help of Riemann curvaturetensor, Ricci tensor and metric tensor.It has been observed that different combinations of Ricci tensor and metric tensor in the defined tensors lead to some of the different geometrical and physical properties.M.S.C. 2010:53C25, 53C43.

On the Geometry of the Riemannian Curvature Tensor of Nearly Trans-Sasakian Manifolds

This paper presents the results of fundamental research into the geometry of the Riemannian curvature tensor of nearly trans-Sasakian manifolds. The components of the Riemannian curvature tensor on the space of the associated G-structure are counted, and the components of the Ricci tensor are calculated. Some identities are obtained that are satisfied by the Riemannian curvature tensors and the Ricci tensor. A number of properties are proved that characterize nearly trans-Sasakian manifolds with a closed contact form. The structure of nearly trans-Sasakian manifolds with a closed contact form is obtained. Several classes are singled out in terms of second-order differential-geometric invariants, and their local structure is obtained. The k-nullity distribution of a nearly trans-Sasakian manifold is studied.

Cosmological Bounces Induced by a Fermion Condensate.

We show that it is possible for fermion condensation of the Nambu-Jona-Lasinio type to induce a nonsingular bounce that smoothly connects a phase of slow contraction to a phase of expansion. A chiral condensate-a nonzero vacuum expectation value of the spinor bilinear ⟨Ψ[over ¯]Ψ⟩-can form spontaneously after a slow contraction phase smooths and flattens the universe and the Ricci curvature exceeds a critical value. In this approach, a high density of spin-aligned free fermions is not required, which avoids the problem of generating a large anisotropy and initiating chaotic mixmaster behavior during the bounce phase.

Singular Weyl’s law with Ricci curvature bounded below

We establish two surprising types of Weyl’s laws for some compact RCD ⁡ ( K , N ) \operatorname {RCD}(K, N) /Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for RCD ⁡ ( K , N ) \operatorname {RCD}(K,N) spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed in Pan and Wei [Geom. Funct. Anal. 32 (2022), pp. 676–685], showing them isometric to the α \alpha -Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures in Cheeger and Colding [J. Differential Geom. 46 (1997), pp. 406–480] and Kapovitch, Kell, and Ketterer [Math. Z. 301 (2022), pp. 3469–3502].

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.

Projective Collineations in Warped Product Manifolds and (PRS)n Manifolds

The current work first explores projective collineations on pseudo-Riemannian manifolds. Projective collineations, curvature collineations, and Ricci curvature collineations are examined in relation to one another. On warped product manifolds, the projective collineations of the form ζ=ζ1+ζ2 are investigated. We scrutinize various inheritance aspects in projective collineations from warped product manifolds to its factor manifolds. This provides, for example, a partially negative solution to Besse’s problem regarding the existence of Einstein warped product manifolds. Finally, Pseudo-Ricci symmetric space-times admitting projective collineations are investigated.

Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Let g0 be a smooth Riemannian metric on a closed manifold Mn of dimension n≥3. We study the existence of a smooth metric g conformal to g0 whose Schouten tensor Ag satisfies the differential inclusion λ(g−1Ag)∈Γ on Mn, where Γ⊂Rn is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric g1 conformal to g0 satisfying λ(g1−1Ag1)∈Γ′‾ in the viscosity sense on Mn, together with a nondegenerate ellipticity condition, where Γ′=Γ or Γ′ is a cone slightly smaller than Γ. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the sign of a nonlinear eigenvalue for the σ2 operator is a conformal invariant in three dimensions. We also give a generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.

On recurrent Riemannian and Ricci curvatures of Finsler metrics

In this paper, we study a property of Riemannian and Ricci curvatures under which it reproduces itself, namely, recurrent Finsler metrics. We prove that if (M,F) is a recurrent Finsler manifold of non-zero isotropic flag curvature, then F is a Landsberg metric. It follows that Every positively complete 2-dimensional Randers metric is recurrent if and only if it is a Riemannian or locally Minkowskian metric. Next, we study two positive (or negative) projectively related Ricci parallel Finsler metrics on a compact manifold. We show that the projective equivalence is trivial and then the Riemannian curvatures are equal. In the same vein, we explore the class of Ricci-recurrent Randers metrics with closed and conformal form, and show that the related Riemannian metric is Ricci-recurrent if and only if the Randers metric is a Berwald metric. Finally, we find the necessary and sufficient condition under which a Kropina metric be Ricci-recurrent, provided that its one-form is closed and conformal.