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In this manuscript, we define (E, F)-invex set, (E, F)-invex functions, and (E, F)-preinvex functions on Euclidean space. We explore the concepts on the Riemannian manifold. We also detail the fundamental properties of (E, F)-preinvex functions and some examples that illustrate the concepts well. We have established a relation between (E, F)-invex and (E, F)-preinvex functions on the Riemannian manifolds. We introduce the conditions A and define (E, F)-proximal sub-gradient. To explore and demonstrate its applicability to optimization problems, (E, F)-preinvex is utilized. In the last, we establish the points of extrema of a non-smooth (E, F)-preinvex functions on (E, F)-invex subset of the Riemannian manifolds by using (E, F)-proximal sub-gradient.

The goal of this research paper is to investigate curvature inheritance symmetry in conformally flat spacetime. Curvature inheritance symmetry in conformally flat spacetime is shown to be a conformal motion. We have proven that a conformally flat spacetime reduces to Einstein spacetime if admits curvature inheritance symmetry. A few results on conformally flat spacetimes that obey Einstein's field equation with or without a cosmological constant, if admits the curvature inheritance symmetry. The energy-momentum tensor is to be covariantly constant in a 4-dimensional relativistic perfect fluid spacetime which is also conformally flat spacetime, admits curvature inheritance, and obeys Einstein's field equations in the presence of a cosmological constant. Moreover, it is also obtained that such spacetimes with perfect fluid satisfy the the vacuum-like equation of state consecutively it is dark matter. Finally, in the third part of the article, the case compatible with all Theorems from Theorem \ref{Th2.1} to Theorem \ref{Th2.5n} is shown. On the other hand, it has also been emphasized that it is an example of de Sitter spacetime. It has been demonstrated that this spacetime also has a conformal killing vector.

This manuscript introduces the idea of GS-exponential kind of convex functions and some of their algebraic features, and we introduce a new class GS-exponential kind of convex sets. In addition, we describe certain fundamental GS-exponential kind of convex function with characteristics in both the general and the differentiable cases. We establish the sufficient conditions of optimality and offer the proof for unconstrained as well as inequality-constrained programming while considering the assumption of GS-exponential kind of convexity.

It is established that general s-convex functions are a new class of generalized convex functions. In a similar vein, a new class of general s-convex sets is introduced, which are generalizations of s-convex sets. Additionally, certain fundamental characteristics of general s-convex functions are discussed for both general cases and differentiable situations. Aside from that, the general s-convexity is used to define and demonstrate the sufficient criteria for optimality for both unconstrained and inequality-constrained programming.

The paper aims to investigate curvature inheritance symmetry in M-projectively flat spacetimes. It is shown that the curvature inheritance 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 $\xi$, when spacetime admits the conditions of both curvature inheritance symmetry and conformal motion or motion along the vector field $\xi$. Also, we have derived some results for M-projectively flat spacetime with perfect fluid following the Einstein field equations with a cosmological term and admitting the curvature inheritance symmetry along the vector field $\xi$. We have shown that an M-projectively flat perfect fluid spacetime obeying the Einstein field equations with a cosmological term and admitting the curvature inheritance symmetry along a vector field $\xi$ 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.

In this paper, we investigate the connection between the M-projective curvature tensor and other tensors. Also, we obtain the divergence of M-projective curvature tensor. A symmetry of spacetime known as M-projective collineation has been presented, and it has been possible to determine the conditions under which the general relativity spacetimes can admit such collineations.

We define w-invex set, w-preinvex, w-strictly preinvex, w-quasi preinvex, w-strictly quasi preinvex, w-semi-strictly quasi preinvex, and w-pre pseudo-invex functions in this context. And these form a class of real functions, which is the generalization of a family of preinvex functions. Here, we provide a thorough analysis of the core characteristics of these functions, along with numerous examples that help to illustrate the idea. Finally, w-quasi preinvex, w-strictly preinvex, and w-strictly quasi preinvex functions are used to analyze the optimization problems.

A class of real functions, which is the generalization of a family of convex functions, is introduced; in this connection, we have defined $X$-convex, strictly $X$-convex, quasi-$X$-convex, strictly quasi-$X$-convex, and semi-strictly quasi-$X$-convex functions. Moreover, in this paper, we give a detailed study of the fundamental properties of these functions with various examples, supporting the concepts. Finally, the study of optimization problems employs quasi-$X$-convex, semistrictly quasi-$X$-convex, and strictly quasi-$X$-convex functions.

In this paper we have introduced a new symmetry property of spacetime which is named as semiconformal curvature collineation, and its relationship with other known symmetry properties has been established. This new symmetry property of the spacetime has also been studied for non-null and null electromagnetic fields.

A spacetime denotes a pure radiation field if its energy momentum tensor represents a situation in which all the energy is transported in one direction with the speed of light. In 1989, Wils and later in 1997 Ludwig and Edgar studied the physical properties of pure radiation metrics, which are conformally related to a vacuum spacetime. In the present paper we investigate the curvature properties of special type of pure radiation metrics presented by Ludwig and Edgar. It is shown that such a pure radiation spacetime is semisymmetric, Ricci simple, $R$-space by Venzi and its Ricci tensor is Riemann compatible. It is also proved that its conformal curvature 2-forms and Ricci 1-forms are recurrent. We also present a pure radiation type metric and evaluate its curvature properties along with the form of its energy momentum tensor. It is interesting to note that such pure radiation type metric is $Ein(3)$ and 3-quasi-Einstein. We also find out the sufficient conditions for which this metric represents a generalized pp-wave, pure radiation and perfect fluid. Finally we made a comparison between the curvature properties of Ludwig and Edgar's pure radiation metric and pp-wave metrics., Comment: 19 pages

The curvature properties of Robinson-Trautman metric have been investigated. It is shown that Robinson-Trautman metric admits several kinds of pseudosymmetric type structures such as Weyl pseudosymmetric, Ricci pseudosymmetric, pseudosymmetric Weyl conformal curvature tensor etc. Also it is shown that the difference $R\cdot R - Q(S,R)$ is linearly dependent with $Q(g,C)$ but the metric is not Ricci generalized pseudosymmetric. Moreover, it is proved that this metric is Roter type, 2-quasi-Einstein, Ricci tensor is Riemann compatible and its Weyl conformal curvature 2-forms are recurrent. It is also shown that the energy momentum tensor of the metric is pseudosymmetric and the conditions under which such tensor is of Codazzi type and cyclic parallel have been investigated. Finally, we have made a comparison between the curvature properties of Robinson-Trautman metric and Som-Raychaudhuri metric., Comment: The paper contained 20 pages

In the differential geometry of certain F-structures, the role of W-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 W-tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing W-tensor have also been considered. The divergence of W-tensor is studied in detail and it is seen, among other results, that a perfect fluid spacetime with conserved W-tensor represents either an Einstein space or a Friedmann-Robertson-Walker cosmological model., Comment: pages 1-15