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We define and look into the fundamental features of X-log convex, X-log quasi convex, X-log semi-strictly quasi convex, and X-log-pseudo convex functions. This paper provides several examples and arguments in favour of the concept. Also, we develope another X-log convex function as the sum of an X-log convex and X-log affine convex function. Additionally, we define local and global log-minimum. Under one condition, it is possible to demonstrate that the local log-minimum of an X-log convex and X-log quasi-convex function is also the global log-minimum. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to examine spacetimes admitting semiconformal curvature tensor in f (ℛ) modify gravity. The semiconformal flatness of general spacetime and spacetime in f (ℛ) gravity with perfect fluid, has been analyzed. For this consideration, we generate the forms of isotropic pressure p and energy density σ. After that, a few energy conditions are taken into account. Finally, we study the divergence-free semiconformal curvature tensor in f (ℛ) gravity in presence of perfect fluid. We emphasize that for recurrent or bi-recurrent energy–momentum tensor, Ricci tensor of this spacetime is semi-symmetric and consequently, the resulting spacetimes either accomplish inflation or possess fixed isotropic pressure and energy density. [ABSTRACT FROM AUTHOR]

In this manuscript, we define a special type convex function on Euclidean space and explore it on the Riemannian manifold. We also detail the fundamental properties of special type convex functions and some examples that illustrate the idea. Moreover, to demonstrate the application to the problems of optimization, these special type convex functions are used. [ABSTRACT FROM AUTHOR]

The semiconformal curvature tensor has been studied for the spacetime of general relativity. It is shown that the energy-momentum tensor with divergence-free semiconformal curvature tensor is of Codazzi type, as well as the energy-momentum tensor of a spacetime having semi-symmetric semiconformal curvature tensor is also semi-symmetric. The semiconformal curvature tensor has also been expressed in terms of different tensors already known in the literature, and the relationship between their divergences has been established. [ABSTRACT FROM AUTHOR]

The paper aims to investigate curvature inheritance (CI) symmetry in M-projectively flat spacetimes. It is shown that the CI symmetry in M-projectively flat spacetime is a conformal motion. We have proved that M-projective curvature tensor follows the symmetry inheritance property along a vector field ξ , when spacetime admits the conditions of both CI symmetry and conformal motion or motion along the vector field ξ. Also, we have derived some results for M-projectively flat spacetime with perfect fluid following the Einstein field equations (EFEs) with a cosmological term and admitting the CI symmetry along the vector field ξ. We have shown that an M-projectively flat perfect fluid spacetime obeying the EFEs with a cosmological term and admitting the CI symmetry along a vector field ξ is either a vacuum or satisfies the vacuum-like equation of state. We have also shown that such spacetimes with the energy–momentum tensor of an electromagnetic field distribution do not admit any curvature symmetry of general relativity. Finally, an example of M-projectively flat spacetime has been exhibited. [ABSTRACT FROM AUTHOR]

In this paper, we have studied conharmonic curvature inheritance (Conh CI) symmetry and its physical importance on the spacetime of general relativity. Also, we have obtained some new results which are direct consequence of proper conformal symmetries. The necessary conditions for a Conh CI vector to be a conformal Killing vector (CKV) have been obtained and results for establishing its relationship between the other known symmetries of spacetimes in general and Einstein spacetimes have been derived. Conh CI with conharmonic motion has been discussed and it is found that a proper Conh CI can produce a new physically relevant solution for perfect fluid spacetime. For Conh CI with conformal Killing vector in perfect fluid spacetime, the equations of state have also been obtained. [ABSTRACT FROM AUTHOR]

In this article, we study curvature inheritance symmetry in Ricci flat spacetimes. We show that, if Ricci flat spacetimes are not of Petrov type N, and admit curvature inheritance symmetries, then the only existing symmetries are conformal motions. We also prove that the only Ricci flat spacetime that admits a proper curvature inheritance symmetry and is of Petrov type other than N is the flat spacetime. Next, we find that the vacuum p p -waves of Petrov type N if admit curvature inheritance symmetry, then conformal motion implies homothetic motion. [ABSTRACT FROM AUTHOR]

In the present paper, our main objective is to study spacetimes which admit a semiconformal curvature tensor. First, we prove that the energy-momentum tensor with vanishing semiconformal curvature tensor, satisfying Einstein's field equations (with cosmological constant), is covariantly constant. Next, we prove that if in a perfect fluid spacetime with divergence-free semiconformal curvature tensor satisfying Einstein field equations without cosmological constant, has constant pressure and density. Finally, we prove that if the perfect fluid spacetime has vanishing semiconformal curvature tensor satisfying Einstein field equations without cosmological constant, then the spacetime has constant energy density and isotropic pressure, and the perfect fluid always behaves as having a cosmological constant. [ABSTRACT FROM AUTHOR]

The motive of the current article is to study and characterize the geometrical and physical competency of the conharmonic curvature inheritance (Conh CI) symmetry in spacetime. We have established the condition for its relationship with both conformal motion and conharmonic motion in general and Einstein spacetime. From the investigation of the kinematical and dynamical properties of the conformal Killing vector (CKV) with the Conh CI vector admitted by spacetime, it is found that they are quite physically applicable in the theory of general relativity. We obtain results on the symmetry inheritance for physical quantities (μ,ρ,ui,σij,η,qi) of the stress-energy tensor in imperfect fluid, perfect fluid and anisotropic fluid spacetimes. Finally, we prove that the conharmonic curvature tensor of a perfect fluid spacetime will be divergence-free when a Conh CI vector is also a CKV. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "Curvature inheritance symmetry on M-projectively flat spacetimes" published in the International Journal of Geometric Methods in Modern Physics. The authors have noted that the results presented in Section 3 of the paper hold for perfect fluid spacetimes in general, not just M-projectively flat spacetimes. Therefore, the term "M-projectively flat" should be removed from the heading, contents of Section 3, abstract, and introduction. The correct form of Equation (26) is also provided in the correction notice. [Extracted from the article]

The semiconformal curvature tensor and its divergence have been studied for perfect fluid spacetimes. It is seen, apart from other results, that the perfect fluid spacetimes with divergence-free semiconformal curvature tensor either satisfy the vacuum-like equation of state or represent FLRW cosmological model. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study the semiconformal curvature tensor for spacetime of general relativity. We establish the results on Killing and conformally Killing vectors satisfying Einstein's field equations in the case of semiconformally flat spacetimes. We extend the same case for the study of cosmological models with dust and perfect fluid. [ABSTRACT FROM AUTHOR]

In the differential geometry of certain -structures, the role of -curvature tensor is very well known. A detailed study of this tensor has been made on the spacetime of general relativity. The spacetimes satisfying Einstein field equations with vanishing -tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing -tensor have also been considered. The divergence of -tensor is studied in detail and it is seen, among other results, that a perfect fluid spacetime with conserved -tensor represents either an Einstein space or a Friedmann-Robertson-Walker cosmological model. [ABSTRACT FROM AUTHOR]

A detailed study of quasi-conformal curvature tensor for the spacetime of general relativity has been made. The spacetimes satisfying Einstein field equations with vanishing quasi-conformal curvature tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing quasi-conformal curvature tensor have also been considered. The divergence of quasi-conformal curvature tensor is studied in the setting of perfect fluid with the derivation of many physical results. [ABSTRACT FROM AUTHOR]

Relationship between W-curvature tensor and its divergence with that of other curvature tensors has been established. A symmetry of the spacetime named as W-collineation, has been introduced and conditions under which the spacetimes of general relativity may admit such collineations are obtained. [ABSTRACT FROM AUTHOR]