Ricci Curvature Research Articles

Hypersurfaces in spaces of constant curvature satisfying a particular Roter type equation

Let M be a hypersurface isometrically immersed in an (n+1)-dimensional semi-Riemannian space of constant curvature, n>3, such that its shape operator A satisfies A3=ϕA2+ψA+ρId, where ϕ, ψ and ρ are some functions on M and Id is the identity operator. The main result of this paper states that on the set U of all points of M at which the square S2 of the Ricci operator S of M is not a linear combination of S and Id, the Riemann-Christoffel curvature tensor R of M is a linear combination of some Kulkarni-Nomizu products formed by the metric tensor g, the Ricci tensor S and the tensor S2 of M, i.e., the tensor R satisfies on U some Roter type equation. Moreover, the (0,4)-tensor R⋅S is on U a linear combination of some Tachibana tensors formed by the tensors g, S and S2. In particular, if M is a hypersurface isometrically immersed in the (n+1)-dimensional Riemannian space of constant curvature, n>3, with three distinct principal curvatures and the Ricci operator S with three distinct eigenvalues then the Riemann-Christoffel curvature tensor R of M also satisfies a Roter type equation of this kind.

Investigation of Indian stock markets using topological data analysis and geometry-inspired network measures

Financial markets are a quintessential example of complex systems wherein strategic connections arising through the trading (buying and selling) of different stocks gives rise to dynamic correlation patterns in stock prices. Network science based approaches have been employed for more than two decades to investigate the stock market as an inferential network, with nodes representing stocks and edges as pairwise cross-correlations between stock prices. However, a recent focus has been the characterization of the higher-order structure, beyond pairwise correlation, of such networks. To this end, geometry-inspired measures (such as discrete Ricci curvatures) and topological data analysis (TDA) based methods (such as persistent homology) have become attractive tools for characterizing the higher-order structure of networks representing the financial systems. In this study, our goal is to perform a comparative analysis of both these approaches, especially by assessing the fragility and systemic risk in the Indian stock markets, which is known for its high volatility and risk. To achieve this goal, we analyze the time series of daily log-returns of stocks comprising the National Stock Exchange (NSE) and the Bombay Stock Exchange (BSE). Specifically, our aim is to monitor the changes in standard network measures, edge-centric discrete Ricci curvatures, and persistent homology based topological measures computed from cross-correlation matrices of stocks. In this study, the edge-centric discrete Ricci curvatures have been employed for the first time in the analysis of the Indian stock markets. The Indian stock markets are known to be less diverse in comparison to the US market, and hence provides us an interesting example. Our results point that, among the persistent homology based topological measures, persistent entropy is simple and more robust than L1-norm and L2-norm of persistence landscape. In a broader comparison between network analysis and TDA, we highlight that the network analysis is sensitive to the way of constructing the networks (threshold or minimum spanning tree), as well as the threshold values used to construct the correlation-based threshold networks. On the other hand, the persistent homology is a more robust approach and is able to capture the higher-order interactions and eliminate noisy data in financial systems, since it does not take into account a single value of threshold but rather a range of values.

Hydrodynamical dual of the gravitational axial anomaly and the cosmological constant

We construct the hydrodynamic expansion for a rotating and accelerated medium in a curved space-time, and establish the relationship between the currents generated by the cosmological constant and the acceleration. Then we consider the more general case with a nonzero Weyl tensor, and show the duality between the current in flat space-time and the gravitational axial anomaly. This generalizes the previous derivation to the case with a nonzero Ricci tensor. Published by the American Physical Society 2024

Ricci Curvatures on Hypersurfaces of Almost Product-like Statistical Manifolds

Riemannian curvature invariants on hypersurfaces of an almost product-like manifold with constant curvature υ are computed. Various relationships involving sectional curvatures and Ricci curvatures have been obtained. Using the Chen-Ricci inequality, some characterizations are presented.

Chern flat, Chern–Ricci flat twisted product in almost Hermitian geometry

Abstract We study Chern flat, Chern–Ricci flat twisted product almost Hermitian manifolds. We extend the twisted product to almost Hermitian manifolds, and give some formulae of the Chern curvature, the Chern–Ricci curvature and the Chern scalar curvature on the twisted product almost Hermitian manifold under the quasi-Kähler condition.

Incompatibility-driven growth and size control during development

Size regulation in living organisms is a major unsolved problem in developmental biology. This is due to the intrinsic complexity of biological growth, which simultaneously involves genetic, biochemical, and mechanical factors. In this article, we propose a novel theoretical framework that explores the role of incompatibility, the geometric source of residual stress in a growing body, as a possible regulator of size termination during development. We explore this paradigm both at the level of a model 2D cell, and at the level of continuous tissues. After establishing a parallel between incompatibility and the shape parameter of vertex models, we show that incompatibility-driven growth leads to size control in a model 2D cell. We then extend the same paradigm to the level of continuous bodies, where incompatibility is measured by the Ricci curvature of the growth tensor. By using the model 2D cell as a template, we now derive an evolutionary law for the growth tensor with curvature fixed at a physiological value. When the analysis is specialised to radial symmetry (discs and spheres), this model captures the salient features observed in Drosophila wing discs and multicellular spheroids: these systems have a target size and build up residual stresses that cause the tissue to open in response to a radial cut, with the cut edges curling outward. The theory proposed in this work suggests that incompatibility in a growing biological tissue is potentially controllable at the cell level, and that incompatibility-driven growth provides an effective method of controlling global information (stress, size) through local geometric controls.

Revisiting Gradient Bach Solitons via Maximum Principles

Supposing that the Ricci curvature has an appropriate lower bound and applying suitable maximum principles, we establish triviality results which guarantee that a gradient Bach soliton must be trivial and Bach-flat. Our approach is based on three main cores: convergence to zero at infinity, polynomial volume growth (both related to complete noncompact Riemannian manifolds) and stochastic completeness.

On Wintgen Ideal Submanifolds Satisfying Some Pseudo-symmetry Type Curvature Conditions

Let M be a Wintgen ideal submanifold of dimension n in a real space form Rn+m(k) of dimension (n + m) and of constant curvaturek, n ≥ 4, m ≥ 1. Let g, R, Ricc, g ∧ Ricc and C be the metric tensor, the Riemann-Christoffel curvature tensor, the Ricci tensor, the Kulkarni-Nomizu product of g and Ricc, and the Weyl conformal curvature tensor of M, respectively. In this paper we study Wintgen ideal submanifolds M in real space forms Rn+m(k), n ≥ 4, m ≥ 1, satisfying the following pseudo-symmetry type curvature conditions: (i) the tensors R · C and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent; (ii) the tensors C · R and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent; (iii) the tensors R·C -C ·R and Q(g, R) (resp., Q(g, C), Q(g, g∧Ricc), Q(Ricc, R) or Q(Ricc, g∧Ricc)) are linearly dependent.

[formula omitted] gravity theory in Einstein space background and causality violation

In this paper, we exploration a Petrov type-N vacuum solution to Einstein’s field equations, while incorporating a negative cosmological constant (Λ<0) within the framework of modified gravity theories. This solution intriguingly accommodates closed time-like curves at a particular moment in time, effectively violates the causality condition, thus acts as a time-machine model. A key observation is that the determinant of the Ricci tensor Rμν for this particular Einstein space metric differs from zero. This noteworthy finding suggests to the existence of an anti-curvature tensor defined Aμν=Rμν−1 and hence, an anti-curvature scalar A=gμνAμν, which is introduced with the Lagrangian of the system, thereby giving rise to as Ricci-inverse gravity theory. We consider class-I models of Ricci-inverse gravity, where the function f=f(R,A)=(R+κA) with κ is the coupling constant. We demonstrate that this Einstein space metric serves as a vacuum solution with a negative modified cosmological constant within the framework of Ricci-inverse gravity. Consequently, the violation of causality persists within this new gravity theory as well. Moreover, we solve the modified field equations by considering matter content other than vacuum and demonstrate that the energy-density and isotropic pressure satisfies the equation ρ=−p=3κΛ.

Chen–Ricci inequality for anti-invariant Riemannian submersions from conformal Kenmotsu space form

AbstractThe aim of this paper is twofold: first, we obtain various curvature inequalities which involve the Ricci and scalar curvatures of horizontal and vertical distributions of anti-invariant Riemannian submersion defined from conformal Kenmotsu space form onto a Riemannian manifold. Second, we obtain the Chen–Ricci inequality for the said Riemannian submersion. The equality cases of all the inequalities are studied. Moreover, these curvature inequalities are studied under two different cases: the structure vector field $$\xi $$ ξ being vertical or horizontal.

Bottom spectrum of three-dimensional manifolds with scalar curvature lower bound

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result for three-dimensional complete manifolds with scalar curvature lower bound subject to some necessary topological assumptions. The rigidity issue is also addressed and a splitting theorem is obtained for such manifolds with the maximal bottom spectrum.

On almost quotient Yamabe solitons

Abstract In this paper, we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call almost quotient Yamabe solitons as they extend quite naturally those already called quotient Yamabe solitons. We present sufficient conditions for a compact almost quotient Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact almost gradient quotient Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.

Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces

Abstract In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.

$L^{p}$ gradient estimates and Calderón–Zygmund inequalities under Ricci lower bounds

In this paper, we investigate the validity of first and second order L^{p} estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present L^{p} estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense, and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain L^{p} estimates for the second order Riesz transform (or, equivalently, the validity of L^{p} Calderón–Zygmund inequalities) on the whole scale 1&lt;p&lt;+\infty by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded, or non-negative in a global integral sense. When 1&lt;p \leq 2 , analogous L^{p} bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order.

Positive Ricci Curvature and the Length of a Shortest Periodic Geodesic

Positive Ricci Curvature and the Length of a Shortest Periodic Geodesic

Space-only gradient estimates of Schrödinger equation with Neumann boundary condition under integral Ricci curvature bounds

Assume that M (M⊆N) is an n-dimensional compact Riemannian submanifold with boundary, satisfying the integral Ricci curvature assumption:D2supx∈N⁡(∮B(x,d)|Ric−|pdy)1p<κ for κ=κ(n,p) small enough, p>n/2, where diam(M)≤D. The boundary of M needs to satisfy the interior rolling R-ball condition. We prove a Hamilton type gradient estimate for positive solutions to the parabolic Schrödinger equation with Neumann boundary conditions, on a compact Riemannian submanifold M with boundary ∂M, satisfying the integral Ricci curvature assumption.

Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with $${\varepsilon }$$-range

Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with $${\varepsilon }$$-range

Lower bounds for the first eigenvalue of Laplacian on graphs

We establish a lower bound for the first eigenvalue of the Dirichlet Laplacian on locally finite graphs, which extending a previous result. We also provide an improved lower bound for the first eigenvalue for finite graphs with non-negative Ricci curvature.

Projective Tensor of Almost Kahler Manifold

The work from this search is analyze the geometrical characteristics of conharmonic tensor of projectine Almost kahler menifold. we use the projective curvature tensor's flatness properties to determine the projective tensor's Almost kahler manifold compounds. (AK-manifold). And found some components Projective of Almost Kahler. proved that the manifold M is a flat holomorphic sectional tensor of projective Almost Kahlar manifold. Prove that the manifold M has J-invariant Ricci tensor. Proved that (AK-manifold) M is kahler manifold. Finally, there exists relation between Almost Kahlar manifold (AK-manifold) and Lacally confermal Kähler menifold (LCK-manifeld) has been found.

Characterizations of vacuum solutions of f(R) − gravity in space-times admitting Z tensor of Codazzi type

In this work, characterizations of vacuum solutions of f(R)-gravity are established in a space-time whose Z tensor is of Codazzi type. We prove that the associated covector of a (PZS) n space-time is an eigenvector of the Ricci tensor, with an eigenvalue equals zero. Additionally, it satisfies compatibility conditions with both the Riemann and Weyl tensors. It is proved that a (PZS) n space-time satisfying f(R)-gravity vacuum solutions is a generalized Friedmann-Robertson-Walker space-time. If n = 4, it becomes a Friedmann-Robertson-Walker space-time.