О некоторых тензорах 6-мерных уплощающихся эрмитовых подмногообразий алгебры Кэли

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M n , g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that $${L^\frac{n+2}{2}}$$ norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C 0 bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M n × [0, T), the curvature tensor stays uniformly bounded on M n × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.

The main objective of this paper is to investigate the symmetry and pseudosymmetry properties of a point-like global monopole (PGM) spacetime, which is a static and spherically symmetric solution of the Einstein’s field equations. It is shown that PGM spacetime fulfills pseudosymmetry due to Weyl conformal (resp., concircular, conharmonic) curvature tensor. It also obeys Ricci generalized conformal pseudosymmetry due to projective curvature tensor and it is Ricci generalized projective pseudosymmetric. Moreover, it is proved that PGM spacetime is [Formula: see text]-quasi Einstein, generalized quasi-Einstein, Einstein manifold of level [Formula: see text] and its Weyl conformal curvature [Formula: see text]-forms are recurrent. The energy–momentum tensor of the PGM spacetime realizes several types of pseudosymmetry, for example, it is pseudosymmetric due to Weyl conformal, concircular and conharmonic curvature tensors. Again both the Ricci tensor and the energy momentum tensor of PGM spacetime are compatible with Riemann, Weyl conformal, projective, conharmonic and concircular curvature tensors. Furthermore, it is exhibited that PGM spacetime reveals motion, curvature collineation and Ricci collineation for the non-Killing vector field [Formula: see text] but it neither possesses curvature collineation nor Ricci collineation for the non-Killing vector field [Formula: see text]. Also, it is shown that PGM spacetime admits [Formula: see text]-almost Ricci soliton for the non-Killing vector field [Formula: see text] but not for the non-Killing vector field [Formula: see text]. The notion of curvature inheritance (resp., curvature collineation) for the (1,3)-type curvature tensor and for the (0, 4)-type curvature tensor is analyzed in PGM spacetime and it is noticed that such notions for [Formula: see text] and (0, 4) curvature tensor are not equivalent. Finally, a significant comparison of the geometric properties between PGM spacetime and interior black hole spacetime is provided.

The space of tensors of metric curvature type on a Euclidean vector space carries a two-parameter family of orthogonally invariant commutative nonassociative multiplications invariant with respect to the symmetric bilinear form determined by the metric. For a particular choice of parameters these algebras recover the polarization of the quadratic map on metric curvature tensors that arises in the work of Hamilton on the Ricci flow. Here these algebras are studied as interesting examples of metrized commutative algebras and in low dimensions they are described concretely in terms of nonstandard commutative multiplications on self-adjoint endomorphisms. The algebra of curvature tensors on a 3-dimensional Euclidean vector space is shown isomorphic to an orthogonally invariant deformation of the standard Jordan product on$3 \times 3$symmetric matrices. This algebra is characterized up to isomorphism in terms of purely algebraic properties of its idempotents and the spectra of their multiplication operators. On a vector space of dimension at least 4, the subspace of Weyl (Ricci-flat) curvature tensors is a subalgebra for which the multiplication endomorphisms are trace-free and the Killing type trace-form is a multiple of the nondegenerate invariant metric. This subalgebra is simple when the Euclidean vector space has dimension greater than 4. In the presence of a compatible complex structure, the analogous result is obtained for the subalgebra of Kähler Weyl curvature tensors. It is shown that the anti-self-dual Weyl tensors on a 4-dimensional vector space form a simple 5-dimensional ideal isometrically isomorphic to the trace-free part of the Jordan product on trace-free$3 \times 3$symmetric matrices.

This study investigates the interconnection between Weyl's curvature tensor \(W_{jkh}^i\) and Cartan's second curvature tensor \(P_{jkh}^i\) in the frame of Finsler geometry (or \(F\)-geometry), a broader framework that generalizes Riemannian geometry(or \(R\)-geometry). When describing the curvature characteristics of \(F\)-space which are crucial for simulating a variety of physical events both tensors are crucial. Even though the geometric meanings and physical consequences of these tensors have been thoroughly investigated, their interconnection remains an open area for research. In the present work, we demonstrate that the Weyl's and Cartan's second curvature tensors are connected by a novel set of identities and inequalities that we deduce by examining their algebraic and geometric characteristics. A series of theorems that outline particular circumstances in which the tensors exhibit generalized birecurrent behavior in Finsler spaces (or \(F\)-spaces) are presented. In addition to offering further insight into how these notions are applied in physics, especially in the gravitational field and cosmology, these results are anticipated to improve our knowledge of the curvature structure in \(F\)-spaces and yield interesting findings in the frame of differential geometry and its physical applications.

This paper investigates the behavior of curvature tensors under the Lie derivative. We derive novel relations between various curvature tensors, such as the Riemann curvature tensor, Ricci tensor, and scalar curvature, when subjected to the Lie derivative. Our results provide a deeper understanding of the geometric properties of manifolds and have potential applications in fields such as general relativity and differential geometry. Also, we build upon the definitions for the conformal and conharmonice curvature tensor in generaralized fifth recurrent Finsler space \(GBK-5RF_n\). We study the various relations between above curvature tensors and the Cartan’s third curvature tensor R\(_{jkh}^i\) by Lie-derivative.

This research delves into the intricate realm of Finsler geometry, with a particular focus on curvature tensors. The paper aims to establish novel identities that govern the expansion of these tensors within the broader context of Finsler space. Through rigorous mathematical analysis, we explore the inter relationships between various curvature tensors and their corresponding expansions. The derived identities not only deepen our understanding of the intrinsic geometric properties of Finsler spaces but also offer potential applications in fields such as physics and engineering where Finsler geometry plays a significant role. This work contributes to the ongoing development of Finsler geometry and provides a foundation for future research in this area. The expansion curvature tensor W_ijk^h is an important geometric object in Finsler spaces. It measures the deviation of the geodesic flow from a parallel flow. In this paper, we investigate some identities for the expansion curvature tensor W_ijk^h in Finsler spaces. These identities provide valuable insights into the geometric properties of Finsler spaces and can be used to derive new results in Finsler geometry. We investigate some identities between Weyl Curvature Tensor W_ijk^h and some other curvature tensors.

In this paper, we introduce and investigate a new class of Finsler spaces, termed generalized | -trirecurrent Finsler spaces, which extend the concept of birecurrence to higher-order covariant derivatives.These spaces are characterized by the third-order h-covariant derivative of Weyl's projective curvature tensor , satisfying a specific recurrence condition.We derive and analyze several key characterizations of this class by formulating higher-order derivative relations for the curvature, torsion, and deviation tensors.Furthermore, we prove that every generalized | -birecurrent Finsler space is inherently a generalized | -trirecurrent Finsler space.Additionally, we explore the connection between Weyl's curvature tensor and Cartan's third curvature tensor within this framework, establishing necessary and sufficient conditions for their recurrence behavior.

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with quarter-symmetric connection preserving the generalized Riemannian metric is actually the Kähler manifold. Observing the six linearly independent curvature tensors with respect to the quarter-symmetric connection, we construct tensors that do not depend on the quarter-symmetric connection generator. One of them coincides with the Weyl projective curvature tensor of symmetric metric $g$. Also, we obtain the relations between the Weyl projective curvature tensor and the holomorphically projective curvature tensor. Moreover, we examine the properties of curvature tensors when some tensors are hybrid.

Considering a non-symmetric linear connection and its dual one appear 6 independent curvature tensors [15]. In the present paper we study properties of curvature tensors of semi-symmetric metric connections on a generalized Riemannian manifold. It is shown that if linear connection preserves the symmetric part of the generalized Riemannian metric with semi-symmetric torsion with an appropriate additional condition, then these six curvature tensors can be expressed by linear combination of Weyl conformal curvature tensor, Weyl projective curvature tensor and the concircular curvature tensor. In the case of the curvature tensor , we have proved that if exists a linear connection preserving the symmetric part of the generalized Riemannian metric with semi-symmetric torsion, whose curvature tensor vanishes, it is necessary and sufficient that the Riemannian metric be projectively flat. For tensor we also get a useful result. If exists a linear connection preserving the symmetric part of the generalized Riemannian metric with semi-symmetric torsion, whose curvature tensor and Ricci tensor vanish, it is necessary and sufficient that the Riemannian metric be conformally flat.

A special class of multidimensional three-webs \( {W^\nabla } \) with covariantly constant curvature and torsion tensors is considered, the curvature tensor having the minimal rank. It is proved that there is a subfamily of adapted frames of the web \( {W^\nabla } \) whose torsion tensor components are constant and whose curvature tensor has a unique nonzero component. The structure equations of the webs of this class are found and some of their properties are described.

A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.

A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.

A simple necessary and sufficient condition that a curvature tensor be Riemannian (i.e., that its symmetric connection be metric) is found, in very general conditions, for a four-dimensional space-time manifold. This enables existing uniqueness results for the metric tensor and an associated algebraic procedure for obtaining the metric tensor components from the curvature tensor components (when the curvature tensor is known to be a Riemannian tensor), to be extended to the situation where it is not known a priori whether the curvature tensor has come from a Lorentz metric.

The primary objective of the article is to investigate the symmetry and pseudosymmetry properties of the Reissner-Nordström-de Sitter (briefly, RNdS) spacetime. The secondary aim of the paper is to explore the notion of Ricci solitons in RNdS spacetimes. The study is important due to the conceding of almost Ricci soliton and almost Ricci Yamabe soliton of the RNdS spacetime. The analysis shows that this spacetime satisfies multiple types of symmetric and pseudosymmetric conditions. It is interesting to note that RNdS spacetime revealed pseudosymmetry, Weyl conformal pseudosymmetry, Weyl projective pseudosymmetry, conharmonic pseudosymmetry, and concircular pseudosymmetry. Also the RNdS spacetime is pseudosymmetric due to conformal, projective, conharmonic curvature tensors. Furthermore, in the RNdS spacetime, the second order commutator tensor R · R is linearly dependent on Q(Ric, R) and Q(g, C), and the tensor R · C does not commute with the tensor C · R. The investigation also shows that RNdS spacetime is a Roter type. It is demonstrated that the RNdS spacetime is 2-quasi Einstein and an Ein(2) space with recurrent conformal 2-forms. Riemann compatibility preserved under geodesic mapping and it has a great geometric insight in determining the nature of Pontryagian forms. We derive the general form of the compatible tensors of the RNdS spacetime. The energy-momentum tensor of the RNdS spacetime is shown to be pseudosymmetric, and also the energy momentum tensor is pseudosymmetric due to conformal, conharmonic, concircular and projective curvature tensor. We note that geodesic maps do not necessarily preserve Weyl compatibility. However such kind of compatibility remains invariant under conformal maps. The presence of a Weyl-compatible vector ensures that the Weyl tensor assumes an algebraically special form, and this condition is both necessary and sufficient for the vanishing of its magnetic component. The energy momentum tensor is compatible with projective, conharmonic, concircular, Riemann and conformal curvature. It is shown that the energy momentum tensor of the RNdS spacetime is Weyl compatible and hence the Weyl tensor of a such a spacetime is algebraically special and hence its magnetic component vanishes. Consequently the Weyl tensor in RNdS spacetime is purely electric. We study a generalized notion of curvature inheritance and find that, with respect to the non-Killing vector fields ∂/∂r and ∂/∂θ, the RNdS spacetime does not satisfy such inheritance condition. However, the RNdS spacetime is shown to admit an almost Ricci soliton and an almost η-Ricci Yamabe soliton with respect to the non-Killing vector field ∂/∂r, but with respect to the non-Killing vector field ∂/∂θ the spacetime does not admit such notions. Finally, we present a comparison between the RNdS and Vaidya-Bonner-de Sitter (VBdS) spacetimes.

One of the earliest general relativity calculations performed with the aid of computer algebra was the computation of the Christoffel symbols and the components of the curvature tensor, Weyl tensor, Ricci tensor and Einstein tensor from the components of a given covariant metric tensor 9ij in some coordinate system. These elementary calculations provide a good illustration of the importance of the choice of algorithms for computer algebra programs in general relativity, a consideration that often seems to have been overlooked. Programs for performing the above calculations were usually based on standard formulas given in general relativity or tensor calculus textbooks. An examination of these formulas reveals that there is a variety of ways of computing the curvature tensor for example.