scholarly journals On quasi-Clifford Osserman curvature tensors

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1241-1247
Author(s):  
Vladica Andrejic ◽  
Katarina Lukic
Keyword(s):  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Duality Principle ◽  
Algebraic Curvature ◽  
Curvature Tensors

We consider pseudo-Riemannian generalizations of Osserman, Clifford, and the duality principle properties for algebraic curvature tensors and investigate relations between them. We introduce quasi- Clifford curvature tensors using a generalized Clifford family and show that they are Osserman. This allows us to discover an Osserman curvature tensor that does not satisfy the duality principle. We give some necessary and some sufficient conditions for the total duality principle.

scholarly journals On Lorentzian spaces of constant sectional curvature

2018 ◽  
Vol 103 (117) ◽  
pp. 7-15
Author(s):  
Vladica Andrejic
Keyword(s):  
Sectional Curvature ◽  
Curvature Tensor ◽  
Jacobi Operator ◽  
Constant Sectional Curvature ◽  
Duality Principle ◽  
Lorentzian Manifolds ◽  
Algebraic Curvature Tensor ◽  
Algebraic Curvature ◽  
Curvature Tensors

We investigate Osserman-like conditions for Lorentzian curvature tensors that imply constant sectional curvature. It is known that Osserman (moreover zwei-stein) Lorentzian manifolds have constant sectional curvature. We prove that some generalizations of the Rakic duality principle (Lorentzian totally Jacobi-dual or four-dimensional Lorentzian Jacobi-dual) imply constant sectional curvature. Moreover, any four-dimensional Jacobi-dual algebraic curvature tensor such that the Jacobi operator for some nonnull vector is diagonalizable, is Osserman. Additionally, any Lorentzian algebraic curvature tensor such that the reduced Jacobi operator for all nonnull vectors has a single eigenvalue has a constant sectional curvature.

scholarly journals On semisymmetric connection

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1179-1184
Author(s):  
Ana Velimirovic ◽  
Milan Zlatanovic
Keyword(s):  
Symmetric Space ◽  
Curvature Tensor ◽  
The Other ◽  
Linear Combinations ◽  
Covariant Derivatives ◽  
Linearly Independent ◽  
Curvature Tensors

Using the non-symmetry of a connection, it is possible to introduce four types of covariant derivatives. Based on these derivatives, several types of Ricci?s identities and twelve curvature tensors are obtained. Five of them are linearly independent but the other curvature tensors can be expressed as linear combinations of these five linearly independent curvature tensors and the curvature tensor of the corresponding associated symmetric space. The semisymmetric connection is defined and the properties of two of the five independent curvature tensors are analyzed. In the same manner, the properties for three others curvature tensors may be derived.

Conformai Curvature Tensors

2011 ◽  
Author(s):  
Charles Fefferman ◽  
C. Robin Graham
Keyword(s):  
Normal Form ◽  
Curvature Tensor ◽  
Riemannian Metric ◽  
Conformal Group ◽  
Conformal Change ◽  
Conformal Curvature ◽  
Covariant Derivatives ◽  
Curvature Tensors ◽  
Derivatives Of

This chapter studies conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on tensors. It is assumed throughout this chapter that n ≥ 3.

scholarly journals A NATURAL CONNECTION ON A BASIC CLASS OF RIEMANNIAN PRODUCT MANIFOLDS

2012 ◽  
Vol 09 (07) ◽  
pp. 1250057 ◽  
Author(s):  
DOBRINKA GRIBACHEVA
Keyword(s):  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Flat Connection ◽  
Riemannian Product ◽  
Natural Connection ◽  
Hermitian Geometry ◽  
Locally Conformal Kähler ◽  
Basic Class ◽  
Necessary And Sufficient ◽  
Locally Conformal

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

scholarly journals A note on the structure of algebraic curvature tensors

2004 ◽  
Vol 382 ◽  
pp. 271-277 ◽  
Author(s):  
J.Carlos Dı́az-Ramos ◽  
Eduardo Garcı́a-Rı́o
Keyword(s):  
Algebraic Curvature ◽  
Curvature Tensors

scholarly journals On Sectional Curvature of Metric Connection with Vectorial Torsion

2020 ◽  
pp. 124-127
Author(s):  
E.D. Rodionov ◽  
V.V. Slavsky ◽  
O.P. Khromova
Keyword(s):  
Sectional Curvature ◽  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Civita Connection ◽  
Modern Physics ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
The Difference ◽  
Topology Of Manifolds ◽  
Conformal Deformations

Papers of many mathematicians are devoted to the study of semisymmetric connections or metric connections with vector torsion on Riemannian manifolds. This type of connectivity is one of the three main types discovered by E. Cartan and finds its application in modern physics, geometry, and topology of manifolds. Geodesic lines and the curvature tensor of a given connection were studied by I. Agricola, K. Yano, and other mathematicians. In particular, K. Yano proved an important theorem on the connection of conformal deformations and metric connections with vector torsion. Namely: a Riemannian manifold admits a metric connection with vector torsion and the curvature tensor being equal to zero if and only if it is conformally flat. Although the curvature tensor of a hemisymmetric connection has a smaller number of symmetries compared to the Levi-Civita connection, it is still possible to define the concept of sectional curvature in this case. The question naturally arises about the difference between the sectional curvature of a semisymmetric connection and the sectional curvature of a Levi-Civita connection.This paper is devoted to the study of this issue, and the authors find the necessary and sufficient conditions for the sectional curvature of the semisymmetric connection to coincide with the sectional curvature of the Levi-Civita connection. Non-trivial examples of hemisymmetric connections are constructed when possible.

scholarly journals Conformal curvature tensors in a generalized Riemannian space in Eisenhart sense

2020 ◽  
pp. 34-34
Author(s):  
Ana Velimirovic
Keyword(s):  
Curvature Tensor ◽  
Riemannian Space ◽  
Conformal Transformation ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Generalized Riemannian Space ◽  
Curvature Tensors ◽  
Special Case

In the present paper generalizations of conformal curvature tensor from Riemannian space are given for five independent curvature tensors in generalized Riemannian space (GRN ), i.e. when the basic tensor is non-symmetric. In earlier works of S. Mincic and M. Zlatanovic et al a special case has been investigated, that is the case when in the conformal transformation the torsion remains invariant (equitorsion transformation). In the present paper this condition is not supposed and for that reason the results are more general and new.

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