Ricci Curvature Research Articles

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.

On Almost C(α)-Manifold Satisfying Certain Curvature Conditions

This research article is about the geometry of the almost C(α)- manifold. Some important properties of the almost C(α)- manifold with respect to the W_3- curvature tensor, such as W_3-flat and W_3- semi-symmetry, are investigated. The relationship of W_3- curvature tensor with Riemann, Ricci, projective, concircular and quasi-conformal curvature tensor is discussed on the almost C(α)- manifold and many important results are obtained. In addition, W_3- pseudo symmetry and W_3- Ricci pseudo symmetry are investigated for the almost C(α)- manifold. The results obtained are interesting and give an idea about the geometry of the almost C(α)- manifold.

The upper bound of the harmonic mean of the Neumann eigenvalues in curved spaces

Considering an n-dimensional Riemannian manifold M whose sectional curvature is bounded above by κ and Ricci curvature is bounded below by (n−1)K, we obtain an upper bound for the harmonic mean of the first (n−1) non-zero Neumann eigenvalues of the Laplacian for domains contained in M. This can be viewed as certain isoperimetric inequality and generalizes the results for domains in space forms [5,15].

η-Ricci--Yamabe and *-η-Ricci--Yamabe solitons in Lorentzian para-Kenmotsu manifolds

Abstract The main purpose of this paper is to study η-Ricci–Yamabe solitons (η-RYS) and * {*} -η-Ricci–Yamabe solitons ( * {*} -η-RYS) in Lorentzian para-Kenmotsu n-manifolds (briefly, ( LPK ) n {(\mathrm{LPK})_{n}} ). We study the curvature condition R . S = 0 {R.S=0} and the cyclic parallel Ricci tensor in ( LPK ) n {(\mathrm{LPK})_{n}} admitting η-RYS. Furthermore, we study M-projectively flat and quasi-M-projectively flat Lorentzian para-Kenmotsu manifolds admitting * {*} -η-RYS. Finally, we give two examples of Lorentzian para-Kenmotsu manifolds admitting η-RYS and * {*} -η-RYS to verify some of our results.

A new approach from the calculus of variations to the Lichnerowicz theorem

The well-known Lichnerowicz theorem states that on an n-dimensional compact Riemannian manifold without boundary with the assumptionRicg≥(n−1)g on the Ricci curvature, the first eigenvalue λ of the Laplacian on the manifold satisfiesλ≥n. This theorem is proved using a Bochner formula.In this paper we give a new approach from the calculus of variations to the Lichnerowicz theorem. We use the energy Econf() of conformality, introduced in [3]. For a smooth map f between Riemannian manifolds, the map f is a weakly conformal map (almost everywhere) if and only if Econf(f) = 0 (the minimum value).Our approach is as follows:(1)The identity map on M is a conformal map, hence a minimizer of the energy Econf(). Especially the identity map is stable, i.e., the second variation is non-negative for the identity map.(2)For the first eigenfunction φ, we take the variation vector field corresponding to dφ, and the non-negativity of the second variation for the variation vector field at the identity map implies the Lichnerowicz theorem.

The Log-Sobolev Inequality for a Submanifold in Manifolds With Asymptotic Non-Negative Intermediate Ricci Curvature

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong et al. (Acta Math. Sci. Ser. B (Engl. Ed.), 44(1):189–194 (2024)) which already generalizes Yi and Zheng (To appear in Chin. Ann. Math., (2023)) and Brendle (Comm. Pure Appl. Math. 75(3), 449–454 (2022)).

Ricci solitons and curvature inheritance on Robinson–Trautman spacetimes

The purpose of this paper is to investigate the existence of Ricci solitons and the nature of curvature inheritance as well as collineations on the Robinson–Trautman (briefly, RT) spacetime. It is shown that under certain conditions RT spacetime admits almost-Ricci soliton, almost-[Formula: see text]-Ricci soliton, almost-gradient [Formula: see text]-Ricci soliton. As a generalization of curvature inheritance [K. L. Duggal, Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times, J. Math. Phys. 33(9) (1992) 2989–2997] and curvature collineation [G. H. Katzin, J. Livine and W. R. Davis, Curvature collineations: A fundamental symmetry property of the space-times of general relativity defined by the vanishing Lie derivative of the Riemann curvature tensor, J. Math. Phys. 10(4) (1969) 617–629], in this paper, we introduce the notion of generalized curvature inheritance and examine if RT spacetime admits such a notion. It is shown that RT spacetime also realizes the generalized curvature (resp., Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) inheritance. Finally, several conditions are obtained, under which RT spacetime possesses curvature (resp., Ricci, conharmonic, Weyl projective) inheritance as well as curvature (resp., Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) collineation, and we have also introduced the concept of generalized Lie inheritance and showed that RT spacetime realizes such a notion.

Some inequalities for bi-slant Riemannian submersions in complex space forms

The goal of this paper is to analyze sharp-type inequalities including the scalar and Ricci curvatures of bi-slant Riemannian submersions in complex space forms. Then, for bi-slant Riemannian submersion between a complex space form and a Riemannian manifold, we give inequalities involving the Casorati curvature of the space ker [Formula: see text]. Also, we mention some examples.

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

In this paper, we examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold M exists for all time t and uniformly converges to a solution to the Yamabe problem on M as t→∞. We show that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on M in the infinite time. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

Positive Intermediate Ricci Curvature with Maximal Symmetry Rank

Positive Intermediate Ricci Curvature with Maximal Symmetry Rank

Slow-roll inflation in f(R,T,RabTab) gravity

In the framework of [Formula: see text] gravity theory, the slow-roll approximation of the cosmic inflation is investigated, where T is the trace of the energy–momentum tensor [Formula: see text], R and [Formula: see text] are the Ricci scalar and tensor, respectively. After obtaining the equations of motion of the gravitational field from the action principle in the spatially flat FLRW metric, the fundamental equations of this theory are received by introducing the inflation scalar field as the matter and taking into account only the minimum curvature-inflation coupling term. Remarkably, after taking the slow-roll approximation, the identical equations as in [Formula: see text] gravity with a [Formula: see text] mixing term are derived. We study several potentials of interest in different domains. We perform analytical analyzes under various approximate conditions, and present numerical results and their comparison with existing observational data at the same time. In the appendix, we analyze the behavior of the inflation scalar field under perturbation while ignoring the effect of metric perturbations. This research complements the slow-roll inflation in the modified theory of gravity.

Chen inequality for general warped product submanifold of Riemannian warped products I×fSm(c)

In the present paper, we investigate the geometry and topology of warped product submanifolds in Riemannian warped product I×fSm(c) and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants (δ-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Our main object is to apply geometrically to number structures and obtained applications of Dirichlet eigenvalues problems. Some new results on mean curvature vanishing are presented, and a partial solution can be found to the well-known problem given by S S Chern. The family of Riemannian warped products generalized to Robertson-Walker space-times

Charting cellular differentiation trajectories with Ricci flow

Complex biological processes, such as cellular differentiation, require intricate rewiring of intra-cellular signalling networks. Previous characterisations revealed a raised network entropy underlies less differentiated and malignant cell states. A connection between entropy and Ricci curvature led to applications of discrete curvatures to biological networks. However, predicting dynamic biological network rewiring remains an open problem. Here we apply Ricci curvature and Ricci flow to biological network rewiring. By investigating the relationship between network entropy and Forman-Ricci curvature, theoretically and empirically on single-cell RNA-sequencing data, we demonstrate that the two measures do not always positively correlate, as previously suggested, and provide complementary rather than interchangeable information. We next employ Ricci flow to derive network rewiring trajectories from stem cells to differentiated cells, accurately predicting true intermediate time points in gene expression time courses. In summary, we present a differential geometry toolkit for understanding dynamic network rewiring during cellular differentiation and cancer.

A quantitative second order estimate for (weighted) p-harmonic functions in manifolds under curvature-dimension condition

We build up a quantitative second-order Sobolev estimate of ln⁡w for positive p-harmonic functions w in Riemannian manifolds under Ricci curvature bounded from below and also for positive weighted p-harmonic functions w in weighted manifolds under the Bakry-Émery curvature-dimension condition.

Classification of the conformally flat Tchebychev affine Kähler hypersurfaces

Tchebychev affine Kähler hypersurfaces were investigated firstly by Xu and Li, who classified the case of dimension 2. As for the case of higher dimension, Xu-Li also proved a classification for the Tchebychev affine Kähler hypersurfaces with complete Calabi metric and nonnegative Ricci curvature. In this paper, the classification of Tchebychev affine Kähler hypersurfaces is obtained under the assumption that Weyl curvature tensor vanishes. As corollaries, we completely determine 3-dimensional Tchebychev affine Kähler hypersurfaces in local sense and obtain the classification of Calabi complete Tchebychev affine Kähler hypersurfaces in R4, which partially improves the results of Xu-Li. In addition, a Bernstein-type result is got for the entire Tchebychev affine Kähler hypersurface in R4.

Perpetual cutoff method and discrete Ricci curvature bounds with exceptions

AbstractOne of the main obstacles regarding Bakry–Emery curvature on graphs is that the results require a global uniform lower curvature bounds where no exception sets are allowed. We overcome this obstacle by introducing the perpetual cutoff method. As applications, we prove gradient estimates only requiring curvature bounds on parts of the graph. Moreover, we sharply upper bound the distance to the exception set for graphs having uniformly positive Bakry–Emery curvature everywhere but on the exception set.

Upper bounds for the critical values of homology classes of loops

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected n-dimensional manifold of positive Ricci curvature Ric≥n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Ric}\\ge n-1$$\\end{document} has length ≤nπ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le n \\pi .$$\\end{document} This improves the bound 8π(n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi (n-1)$$\\end{document} given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).

On the absence of ghosts in quadratic bigravity

An extension of the bimetric theory of gravity is considered that includes quadratic Ricci curvature terms associated with each metric. The issue of the Boulware-Deser ghost is analyzed. The Hamiltonian constraint is derived and the existence of a secondary constraint is shown, proving that the theory is ghost-free.

Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature R i c N \mathrm {Ric}_N is bounded from below by a real number K K in every timelike direction satisfies the timelike curvature-dimension condition T C D q ( K , N ) \mathrm {TCD}_q(K,N) for all q ∈ ( 0 , 1 ) q\in (0,1) . The converse and a nonpositive-dimensional version ( N ≤ 0 N \le 0 ) of this result are also shown. Our discussion is based on the solvability of the Monge problem with respect to the q q -Lorentz–Wasserstein distance as well as the characterization of q q -geodesics of probability measures. One consequence of our work is the sharp timelike Brunn–Minkowski inequality in the Lorentz–Finsler case.

Diameter estimates in Kähler geometry

AbstractDiameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for estimates for the Monge–Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, we solve the long‐standing problem of uniform diameter bounds and Gromov–Hausdorff convergence of the Kähler–Ricci flow, for both finite‐time and long‐time solutions.