Scalar Curvature Function Research Articles

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.

Non-commutative wormhole in non-minimal curvature–matter coupling of f(R) gravity with Gaussian and Lorentzian distributions

In this work, we construct two new wormhole solutions in the theory dealing with non-minimal coupling between curvature and matter. We take into account an explicitly non-minimal coupling between an arbitrary function of scalar curvature [Formula: see text] and the Lagrangian density of matter. For this purpose, we discuss the Wormhole geometries inspired by non-minimal curvature coupling in [Formula: see text] gravity for linear model in [Formula: see text] as well as nonlinear model in [Formula: see text]. To derive these solutions, we choose the Gaussian and Lorentzian density distributions. To check the viability of these solutions, we plot the graphs for energy conditions and wormhole parameters. It is found that obtained wormhole solutions in both distributions satisfy the energy condition. The resulting wormhole solutions for both non-commutative distributions are determined to be physically stable when we evaluate the stability of these wormhole solutions graphically. It is concluded that wormhole solutions exist with viable physical properties in the non-minimal curvature–matter coupling of [Formula: see text] gravity with Gaussian and Lorentzian distributions.

Static Spherically Symmetric Spacetime as a Function of Scalar Curvature

Static Spherically Symmetric Spacetime as a Function of Scalar Curvature

Prescribed Webster Scalar Curvatures on Compact Pseudo-Hermitian Manifolds

In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.

The rigidity of Ricci shrinkers of dimension four

In dimension $4$, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinkers satisfying some natural geometric properties.

The formulation of f(R)-gravity on singular semi-Riemannian manifolds

In singular semi-Riemannian manifolds, the metric is allowed to be degenerate, i.e. it is not necessarily non-degenerate. The formulation of f(R)-gravity on this kind of manifold has been studied. The field equation is obtained via variational principle approach. The quantities which is involved in the formulation is a smooth quantity even when the metric becomes degenerate. These quantities are the densitized energy-momentum tensor, the Ricci curvature tensor, and the arbitrary function of scalar curvature. The resulting filed equation is a more general field equation. It reduces to the standard field equation when the metric is non-degenerate. The extension of the modified Friedmann equations has also been obtained. The generalized mass (energy) density and the generalized pressure density can be constructed based on the definition of the densitized energy-momentum tensor. These quantities remain smooth even when the metric is degenerate. The modified-f(R) Schwarzschild black holes has also been considered. By coordinate transformation, the metric which is semi-regular has been obtained. The consequences of this formulation to the spacetime singularity has also been discussed.

Yamabe flow with prescribed scalar curvature

In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.

Sharp Isoperimetric Inequalities for Small Volumes in Complete Noncompact Riemannian Manifolds of Bounded Geometry Involving the Scalar Curvature

Abstract We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with $C^3$ bounded geometry in a suitable sense involving the scalar curvature function. Under $C^3$ bounds of the geometry, if the supremum of scalar curvature function $S_g<n(n-1)k_0$ for some $k_0\in \mathbb{R}$, then for small volumes the isoperimetric profile of $(M^n,g)$ is less then or equal to the isoperimetric profile of the complete simply connected space form of constant sectional curvature $k_0$. This work generalizes Theorem $2$ of [12] in which the same result was proved in the case where $(M^n, g)$ is assumed to be compact. As a consequence of our result we give an asymptotic expansion in Puiseux series up to the 2nd nontrivial term of the isoperimetric profile function for small volumes, generalizing our earlier asymptotic expansion [29]. Finally, as a corollary of our isoperimetric comparison result, it is shown that for small volumes the Aubin–Cartan–Hadamard’s conjecture is true in any dimension $n$ in the special case of manifolds with $C^3$ bounded geometry, and $S_g<n(n-1)k_0$. Two different intrinsic proofs of the fact that an isoperimetric region of small volume is of small diameter. The 1st under the assumption of mild bounded geometry, that is, positive injectivity radius and Ricci curvature bounded below. The 2nd assuming the existence of an upper bound of the sectional curvature, positive injectivity radius, and a lower bound of the Ricci curvature.

Refined estimates for simple blow-ups of the scalar curvature equation on $S^n$

In their work on a sharp compactness theorem for the Yamabe problem, Khuri, Marques and Schoen apply a refined blow-up analysis (what we call `second order blow-up argument' in this article) to obtain highly accurate approximate solutions for the Yamabe equation. As for the conformal scalar curvature equation on S^n with n > 3, we examine the second order blow-up argument and obtain refined estimate for a blow-up sequence near a simple blow-up point. The estimate involves local effect from the Taylor expansion of the scalar curvature function, global effect from other blow-up points, and the balance formula as expressed in the Pohozaev identity in an essential way.

Aspects of Positive Scalar Curvature and Topology I

This is the second and concluding part of a survey article. Whether or not a smooth manifold admits a Riemannian metric whose scalar curvature function is strictly positive is a problem which has been extensively studied by geometers and topologists alike. More recently, attention has shifted to another intriguing problem. Given a smooth manifold which admits metrics of positive scalar curvature, what can we say about the topology of the space of such metrics? We provide a brief survey, aimed at the non-expert, which is intended to provide a gentle introduction to some of the work done on these deep questions.

On the prescribed scalar curvature problem on Sn: Part 1, asymptotic estimates and existence results

There have been many works on the problem of finding a conformal metric on the standard sphere Sn,n≥3, when the prescribed scalar curvature function is flat near its critical points with order of flatness β. All the existence results up to this research concern the case 1<β<n. The present paper deals with the case β≥n. We provide a complete analysis of the asymptotic expansion of the associated gradient vector field for any β>1. Moreover, we establish (in this first part) existence results concerning some cases of β≥n.

Non-existence of Funk functions for Finsler spaces of non-vanishing scalar flag curvature

In his book Differential Geometry of Spray and Finsler spaces, page 177, Zhongmin Shen asks “whether or not there always exist non-trivial Funk functions on a spray space”. In this note, we will prove that the answer is negative for the geodesic spray of a Finslerian function of non-vanishing scalar flag curvature.

Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds

Our purpose in this paper is to study the rigidity of complete linear Weingarten hypersurfaces immersed in a locally symmetric manifold obeying some standard curvature conditions (in particular, in a Riemannian space with constant sectional curvature). Under appropriated constrains on the scalar curvature function, we prove that such a hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of them being simple. Furthermore, we also deal with the parabolicity of these hypersurfaces with respect to a suitable Cheng–Yau modified operator.

THE EXISTENCE OF WARPING FUNCTIONS ON RIEMANNIAN WARPED PRODUCT MANIFOLDS WITH FIBER MANIFOLDS OF CLASS (A)

Abstract. In this paper, we prove the existence and the nonex-istence of warping functions on Riemannian warped product mani-folds with some prescribed scalar curvatures according to the bermanifolds of class (A). 1. IntroductionOne of the basic problems in the di erential geometry is studying theset of curvature functions which a given manifold possesses.The well-known problem in di erential geometry is that of whetherthere exists a warping function of warped metric with some prescribedscalar curvature function. One of the main methods of studying di er-ential geometry is by the existence and the nonexistence of a warpedmetric with prescribed scalar curvature functions on some Riemannianwarped product manifolds. In order to study these kinds of problems,we need some analytic methods in di erential geometry.For Riemannian manifolds, warped products have been useful in pro-ducing examples of spectral behavior, examples of manifolds of negativecurvature (cf. [2, 3, 4, 5, 7, 13, 14]), and also in studying L

Scalar Curvature Functions of Almost-Kähler Metrics

For a closed smooth manifold M admitting a symplectic structure, we define a smooth topological invariant Z(M) using almost-Kahler metrics, i.e., Riemannian metrics compatible with symplectic structures. We also introduce $$Z(M, [[\omega ]])$$ depending on symplectic deformation equivalence class $$[[\omega ]]$$ . We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equivalence class with different signs of $$Z(M, [[\omega ]] )$$ . Using Z invariants, we set up a Kazdan–Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold $$(M, \omega )$$ of dimension $$\ge \!\!4$$ , any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kahler metric compatible with a symplectic form which is deformation equivalent to $$\omega $$ .

The Yamabe problem on stratified spaces

We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of structural assumptions about the space and of the behavior of the scalar curvature function on its smooth locus. The second half of this paper shows how this result applies in the category of smoothly stratified pseudomanifolds, and we also prove sharp regularity for the solutions on these spaces. This sharpens and generalizes the results of Akutagawa and Botvinnik (GAFA 13:259–333, 2003) on the Yamabe problem on spaces with isolated conic singularities.

FINSLER METRIZABLE ISOTROPIC SPRAYS AND HILBERT’S FOURTH PROBLEM

AbstractIt is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In our main result we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. We call these conditions the scalar flag curvature (SFC) test. The proof of the main result provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. Hilbert’s fourth problem asks to determine the Finsler functions with rectilinear geodesics. A Finsler function that is a solution to Hilbert’s fourth problem is necessarily of constant or scalar flag curvature. Therefore, we can use the constant flag curvature (CFC) test, which we developed in our previous paper, Bucataru and Muzsnay [‘Sprays metrizable by Finsler functions of constant flag curvature’, Differential Geom. Appl.31 (3)(2013), 405–415] as well as the SFC test to decide whether or not the projective deformations of a flat spray, which are isotropic, are metrizable by Finsler functions of constant or scalar flag curvature. We show how to use the algorithms provided by the CFC and SFC tests to construct solutions to Hilbert’s fourth problem.

Classification of complete projective special real surfaces

We determine all complete projective special real surfaces. By the supergravity r-map, they give rise to complete projective special Kähler manifolds of dimension 6, which are distinguished by the image of their scalar curvature function. By the supergravity c-map, the latter manifolds define in turn complete quaternionic Kähler manifolds of dimension 16.

SCALAR CURVATURE FUNCTIONS OF ALMOST-KÄHLER METRICS ON A CLOSED SOLV-MANIFOLD

We discuss on the classification problem of symplectic manifolds into three families according to the scalar curvature functions of almost K\{a}hler metrics they admit. We also present a 4-dimensional solv-manifold as an example which belongs to one of the three families.

A NOTE ON SCALAR CURVATURE FUNCTIONS OF ALMOST-KÄHLER METRICS

We present a 4-dimensional nil-manifold as the ?rst example of a closed non-KAahlerian symplectic manifold with the following property: a function is the scalar curvature of some almost KAahler metric i® it is negative somewhere. This is motivated by the Kazdan-Warner`s work on classifying smooth closed manifolds according to the possible scalar curvature functions.