scholarly journals NATURAL CONNECTION WITH TOTALLY SKEW-SYMMETRIC TORSION ON RIEMANNIAN ALMOST PRODUCT MANIFOLDS

2012 ◽  
Vol 09 (01) ◽  
pp. 1250003 ◽  
Author(s):  
DIMITAR MEKEROV ◽  
MANCHO MANEV
Keyword(s):  
Linear Connection ◽  
Riemannian Metric ◽  
Sufficient Condition ◽  
Product Manifold ◽  
Necessary And Sufficient Condition ◽  
Natural Connection ◽  
Almost Product Manifold ◽  
Hermitian Geometry ◽  
Levi Civita Connection ◽  
Necessary And Sufficient

On a Riemannian almost product manifold (M, P, g), we consider a linear connection preserving the almost product structure P and the Riemannian metric g and having a totally skew-symmetric torsion. We determine the class of the manifolds (M, P, g) admitting such a connection and prove that this connection is unique in terms of the covariant derivative of P with respect to the Levi-Civita connection. We find a necessary and sufficient condition the curvature tensor of the considered connection to have similar properties like the ones of the Kähler tensor in Hermitian geometry. We pay attention to the case when the torsion of the connection is parallel. We consider this connection on a Riemannian almost product manifold (G, P, g) constructed by a Lie group G.

scholarly journals Einstein doubly warped product manifolds with semi-symmetric metric connection

2020 ◽  
Vol 57 ◽  
pp. 7-24
Author(s):  
Punam Gupta ◽  
Abdoul Salam Diallo
Keyword(s):  
Warped Product ◽  
Sufficient Condition ◽  
Product Manifold ◽  
Necessary And Sufficient Condition ◽  
Class A ◽  
Warped Product Manifold ◽  
Metric Connection ◽  
Necessary And Sufficient

In this paper, we study the doubly warped product manifolds with semi-symmetric metric connection. We derive the curvature formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for an Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection.

On An Affine Connection Which Admits A Volume-Like Form

1990 ◽  
Vol 33 (4) ◽  
pp. 482-488 ◽  
Author(s):  
D. P. Chi ◽  
Y. D. Yoon
Keyword(s):  
Affine Connection ◽  
Riemannian Metric ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Riemannian Connection ◽  
Cech Cohomology ◽  
Necessary And Sufficient

AbstractA necessary and sufficient condition to obtain a volumelike form from an affine connection is given in terms of the Čech cohomology, after the volume-like form is naturally defined without a Riemannian metric. A necessary condition for an affine connection to be a Riemannian connection for some metric is also given.

scholarly journals On the notion ofL 1-completeness of a stochastic flow on a manifold

2002 ◽  
Vol 7 (12) ◽  
pp. 627-635 ◽  
Author(s):  
Yu. E. Gliklikh ◽  
L. A. Morozova
Keyword(s):  
Functional Space ◽  
Riemannian Metric ◽  
Stochastic Flow ◽  
Time Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
The Given

We introduce the notion ofL 1-completeness for a stochastic flow on manifold and prove a necessary and sufficient condition for a flow to beL 1-complete.L 1-completeness means that the flow is complete (i.e., exists on the given time interval) and that it belongs to some sort ofL 1-functional space, natural for manifolds where no Riemannian metric is specified.

scholarly journals Four-Dimensional Semi-Riemannian Szabó Manifolds

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Abdoul Salam Diallo ◽  
Punam Gupta
Keyword(s):  
Ricci Tensor ◽  
Riemannian Metric ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Affine Surface ◽  
Necessary And Sufficient

In this paper, we prove that the deformed Riemannian extension of any affine Szabó manifold is a Szabó pseudo-Riemannian metric and vice versa. We prove that the Ricci tensor of an affine surface is skew-symmetric and nonzero everywhere if and only if the affine surface is Szabó. We also find the necessary and sufficient condition for the affine Szabó surface to be recurrent. We prove that, for an affine Szabó recurrent surface, the recurrence covector of a recurrence tensor is not locally a gradient.

On Projectively Flat (α, β)-metrics

2009 ◽  
Vol 52 (1) ◽  
pp. 132-144 ◽  
Author(s):  
Zhongmin Shen
Keyword(s):  
Riemannian Metric ◽  
Finsler Metrics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Regular Case ◽  
Projectively Flat ◽  
Necessary And Sufficient

AbstractThe solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called (α, β)-metrics defined by a Riemannian metric α and a 1-form β, and find a necessary and sufficient condition for such metrics to be projectively flat in dimension n ≥ 3.

Projectively Flat Finsler Space of Douglas Type with Weakly-Berwald (α,β)-Metric

2017 ◽  
Vol 18 ◽  
pp. 1-12
Author(s):  
Maranna Ramesha ◽  
S.K. Narasimhamurthy
Keyword(s):  
Present Article ◽  
Finsler Space ◽  
Riemannian Metric ◽  
Sufficient Condition ◽  
Important Class ◽  
Necessary And Sufficient Condition ◽  
Berwald Space ◽  
Projectively Flat ◽  
Necessary And Sufficient

The present article is organized as follows: In the first part, we characterize the important class of special Finsler (α,β)-metric in the form ofL=α+α2/β, whereαis Riemannian metric andβis differential 1-form to be projectively flat. In the second part, we describe condition for a Finsler spaceFnwith an (α,β)-metric is of Douglas type. Further we investigate the necessary and sufficient condition for a Finsler space with an (α,β)-metric to be weakly-Berwald space and Berwald space.

On general (α, β)-metrics with vanishing Douglas curvature

2015 ◽  
Vol 26 (09) ◽  
pp. 1550076 ◽  
Author(s):  
Hongmei Zhu
Keyword(s):  
Riemannian Metric ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Douglas Metric ◽  
Douglas Curvature

In this paper, we study a class of Finsler metrics called general (α, β)-metrics, which are defined by a Riemannian metric α and a 1-form β. We find an equation which is necessary and sufficient condition for such Finsler metric to be a Douglas metric. By solving this equation, we obtain all of general (α, β)-metrics with vanishing Douglas curvature under certain condition. Many new non-trivial examples are explicitly constructed.

scholarly journals UNIFIED CONTACT STRUCTURE

1991 ◽  
Vol 22 (2) ◽  
pp. 125-132
Author(s):  
B. B. SINHA ◽  
R. N. YADAV
Keyword(s):  
Contact Structure ◽  
Linear Connection ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

A necessary and sufficient condition that a $C^\infty$ manifold to admits anunified contad structure is established. A condition is obtained so that it becomes P-Sasakian. A linear connection in it is defined so its distributions are parallel.

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