arXiv research

In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann-Christoffel curvature tensor, the same type structure given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various ge…

The projective curvature tensor $P$ is invariant under a geodesic preserving transformation on a semi-Riemannian manifold. It is well known that $P$ is not a generalized curvature tensor and hence it possesses different geometric properties than other generalized curvature tensors. The main object of the present paper …

We show that every Kaehler algebraic curvature tensor is geometrically realizable by a Kaehler manifold of constant scalar curvature. We also show that every para-Kaehler algebraic curvature tensor is geometrically realizable by a para-Kaehler manifold of constant scalar curvature

We present an algebraic investigation of generalized and equiaffine curvature tensors in a given pseudo-Euclidean vector space and study different orthogonal, irreducible decompositions in analogy to the known decomposition of algebraic curvature tensors. We apply the decomposition results to characterize geometric pro…

The paper examines geometric curvatures in generalized Riemannian spaces.

Classifies weakly Einstein curvature tensors in 4D Euclidean space.

The paper examines geometric properties of a unique spacetime model.

We consider the hyperbolic geometric flow $\frac{\partial^2}{\partial t^2}g(t)=-2Ric_{g(t)}$ introduced by Kong and Liu [KL]. When the Riemannian metric evolve, then so does its curvature. Using the techniques and ideas of S.Brendle [Br,BS], we derive evolution equations for the Levi-Civita connection and the curvature…

Defines a new tensor related to special geometric spaces.

Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If $π$ is a spacelike 2 plane, let $R(π)$ be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hyp…

We give coordinate formula and geometric description of the curvature of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain types with respect to a pair of a linear connection on a vector bundle and a lin…

Study on tautness tensor for Riemannian foliations.

New tensors reveal full curvature structure from Riemann tensor.

New method calculates Ricci curvature from distances between weighted volumes.

We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6…

We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric tor…

Study spectral functionals on manifolds with torsion.

Constructs a unique Levi-Civita connection for generalised metrics.

In this paper we discuss curvature tensors in the context of Absolute Parallelism geometry. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection. Using the Bianchi identities some other identities are derived from the expressions obtained. These identiti…

Proves existence of Yang-Mills fields for specific curvature conditions.

In this paper we provide a \emph{global} investigation of the geometry of parallelizable manifolds (or absolute parallelism geometry) frequently used for application. We discuss the different linear connections and curvature tensors from a global point of view. We give an existence and uniqueness theorem for a remarkab…

The classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent b…

Investigates geometric properties of Bardeen black hole spacetime.

On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manif…

Paper introduces $*-$Ricci flow and its properties.

In Finsler geometry, we use calculus to study the geometry of regular inner metric spaces. In this note I will briefly discuss various curvatures and their geometric meanings from the metric geometry point of view, without going into the forest of tensors.

Derives curvature conditions for spatial isotropy without field equations.

We determine the space of algebraic pseudo-Hermitian Kähler-Weyl curvature tensors and the space of para-Hermitian Kähler-Weyl curvature tensors in dimension 4 and show that every algebraic possibility is geometrically realizable. We establish the Gray identity for pseudo-Hermitian Weyl manifolds and for para-Hermitian…

Study geometric properties and physical applications of mixed quasi-Einstein spacetime.

We develop a comprehensive geometric framework for defining spaces $\mathcal{G}(M,E)$ of nonlinear generalized sections of vector bundles $E \to M$ containing spaces of distributional sections $\mathcal{D}'(M, E)$. Our theory incorporates classical differential geometric operations (like tensor products, covariant deri…

An absolute parallelism (AP-) space having Finslerian properties is called FAP-space. This FAP-structure is more wider than both conventional AP and Finsler structures. In the present work, more geometric objects as curvature and torsion tensors are derived in the context of this structure. Also second order tensors, u…

In this paper, we deal with a generalization of the geometry of parallelizable manifolds, or the absolute parallelism (AP-) geometry, in the context of generalized Lagrange spaces. All geometric objects defined in this geometry are not only functions of the positional argument $x$, but also depend on the directional ar…

We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near …

The study characterizes symmetries in Kaehler manifolds.

Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation …

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ${\mathbb{R}}^{2}$ in certain class…

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we int…

We employ the language of Cartan's geometry to present a model for studying vector spaces of Killing two-tensors defined in pseudo-Riemannian spaces of constant curvature under the action of the corresponding isometry group. We also discuss geometric properties of joint invariants of Killing two-tensors defined in the …

A spacetime is a connected 4-dimensional semi-Riemannian manifold endowed with a metric $g$ with signature $(- + + +)$. The geometry of a spacetime is described by the metric tensor $g$ and the Ricci tensor $S$ of type $(0, 2)$ whereas the energy momentum tensor of type $(0,2)$ describes the physical contents of the sp…

The paper investigates various generalizations of semisymmetric and pseudosymmetric manifolds.

In recent years, Streets and Tian introduced a series of curvature flows to study non-Kähler geometry. In this paper, we study how to construct second order curvature flows in a uniform way, under some natural assumptions which holds in Streets and Tian's works. As a result, by classifying the lower order tensors, we c…

Modified Gibbs-Helmholtz equation geometric models for thermodynamics.

We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hyper…

The paper proves geometric comparisons on metric measure spaces with integral Bakry-Émery Ricci tensor bounds.

The paper explores symmetries in Kähler manifolds using Ricci tensor properties.

The paper studies connections in superintegrable systems, revealing geometric insights.

In this paper, we investigate geometric properties of some curvature tensors of a four-dimensional Walker manifold. Some characterization theorems are also obtained.

New rigidity results for tensors on non-compact manifolds with curvature conditions.