Farah Hasan Jasim AlHusseini

This literature review focuses on six studies of the conharmonic curvature tensor in almost Hermitian and almost contact metric manifolds. The conharmonic curvature tensor is a more effective constant invariant for detecting conformal curvature events than traditional invariants. The principal methodological concern is the absence of a thorough investigation that integrates previous research outcomes regarding the conharmonic curvature tensor. Conformal flatness and its related invariants impose significant structural constraints on the Vaisman-Gray, nearly Kähler, and locally conformal cosymplectic manifolds, frequently leading to Einstein or Einstein-normalized metrics or a reduction to locally conformal Kähler representations. A nearly conformal Kähler structure with a vanishing conharmonic tensor has been established, and harmonic flattening in dimensions more than four leads to a locally conformal Kähler structure with vanishing scalar curvature in the Vaisman-Gray case. These manifolds are proven to be equivalent to Einstein manifolds under additional assumptions, and necessary and sufficient conditions for the existence of point-homogeneous segmental tensors are specified. Three harmonic constants that determine the various 𝜂-Einstein classes of type (𝛼,𝛽) have been established in the normal state of a flat conharmonic tensor, whereas locally conformal manifolds must satisfy harmonic flatness, normality of distribution, and 𝜂-Einstein's condition. A classification based on harmonic behavior is constructed in a nearly locally identical space; paracontact criteria are defined, and manifolds are shown to be homogeneous paracontact. Finally, we present the main conclusion derived from the results of this literature review, along with an illustrative application. The findings indicate that conharmonic curvature tensor helps establish novel invariant-based classifications for future studies in differential geometry.

This paper focuses on the essential role of the quasi-conformal curvature tensor in developing the theoretical foundation and applications of quasi-contact metric geometry. The quasi-conformal curvature tensor of NC 10 -manifolds using the space associated with G structures (AGS-space) has been investigated. By deriving explicit expressions for the tensor components, the necessary and sufficient conditions for considering the quasi-conformal NC 10 -manifold as flat quasi-conformal have been determined. Noteworthy, it indicated that any flat quasi-conformal NC 10 -manifold is locally isometric to the product of the Kähler manifold and the real line, thus connecting abstract tensor properties to classical geometric structures. Moreover, the conditions for the NC 10 -manifold that yield the η -Einstein structure are established. New analogues of Gray’s identities for the quasi-conformal curvature tensor have been introduced for NC 10 -manifold.

This study aims to investigate the quasi-conformal curvature tensor of NC10-manifold. The components of this tensor were determined using the adjoined G-structure space. Three quasi-conformal invariants were identified in relation to the vanishing quasi-conformal curvature tensor. Subsequently, three types of NC10-manifold were established. Furthermore, the necessary conditions for these classes to be an η-Einstein manifold were established.

Abstract: The authors classified a locally conformal almost cosympleсtic manifold ( L C A C S -manifold) according to the conharmonic curvature tensor. In particular, they have determined the necessary conditions for a conharmonic curvature tensor on the L C A C S -manifold of classes C T i , i = 1 , 2 , 3 to be Φ -quaisi invariant. Moreover, it has been proved that any L C A C S -manifold of the class C T 1 is conharmoniclly Φ -paracontact.

The main purpose of the present paper is to study the geometric properties of the conharmonic curvature tensor of normal locally conformal almost cosymplectic manifolds (normal LCAC-manifold). In particular, three conhoronic invariants are distinguished with regard to the vanishing conharmonic tensor. Subsequentaly, three classes of normal LCAC-manifolds are established. Moreover, it is proved that the manifolds of these classes are $ \eta $-Einstein manifolds of type $ (\alpha,\beta) $. Furthermore, we have determined $ \alpha $ and $ \beta $ for each class.

This paper aims to study the geometrical properties of the conharmonic curvature tensor of a locally conformal almost cosymplectic manifold. The necessary and sufficient conditions for the conharmonic curvature tensor to be flat, the locally conformal almost cosymplectic manifold to be normal and an η -Einstein manifold were determined.

In this article, we introduced the concept of holomorphic sectional conharmonic tensor of a normal local conformally almost co-symplectic manifold ( -manifold). In particular, we established some of its properties and analytical expressions. Consequently, analytical conditions for the -manifold to be a kind of point wise constancy conharmoniclly holomorphic sectional are obtained.

The purpose of the present paper is to discuss the geometrical properties of a locally conformal almost cosymplectic manifold of constant curvature. In particular, the necessary and sufficient conditions for the aforementioned manifold to be of constant curvature have been determined.

The aim of the present paper is to study the geometry of locally conformal almost cosymplectic manifold of -holomorphic sectional conharmonic curvature tensor. In particular, the necessary and sucient conditions that locally conformal almost cosymplectic manifold is a manifold of point constant -holomorphic sectional conharmonic curvature tensor have been found. The relation between the mentioned manifold and the Einstein manifold is determined.