On generalized concircularly recurrent manifolds

2009 ◽  
Vol 46 (2) ◽  
pp. 287-296
Author(s):  
U. De ◽  
A. Kalam Gazi
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Einstein Manifold ◽  
Constant Scalar Curvature ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Recurrent Manifold ◽  
New Type ◽  
Necessary And Sufficient

In this paper we study a new type of Riemannian manifold called generalized concircularly recurrent manifold. We obtain a necessary and sufficient condition for the constant scalar curvature of such a manifold. Next we study Ricci symmetric generalized concircularly recurrent manifold and prove that such a manifold is an Einstein manifold. Finally, we obtain a sufficient condition for a generalized concircularly recurrent manifold to be a special quasi-Einstein manifold.

scholarly journals On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1941
Author(s):  
Sharief Deshmukh ◽  
Nasser Bin Turki ◽  
Haila Alodan
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Rigidity Results ◽  
Necessary And Sufficient

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.

scholarly journals A Characterization of Invariant Affine Connections

1960 ◽  
Vol 16 ◽  
pp. 35-50 ◽  
Author(s):  
Bertram Kostant
Keyword(s):  
Riemannian Manifold ◽  
Simple Proof ◽  
General Theorem ◽  
Transitive Group ◽  
Complete Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Simply Connected

1. Introduction and statement of theorem. 1. In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem — Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the Tx of [1] is just the ax of [6] when X is restricted to p0, see [6], p. 539).

TOTALLY GEODESIC SUBMANIFOLDS OF GL-MANIFOLDS

2012 ◽  
Vol 09 (01) ◽  
pp. 1250002
Author(s):  
ABOLGHASEM LALEH ◽  
MORTEZA M. REZAII ◽  
ATAABAK BAAGHERZADEH HUSHMANDI
Keyword(s):  
Riemannian Manifold ◽  
Finsler Manifold ◽  
Finsler Metric ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Totally Geodesic ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Necessary And Sufficient ◽  
Sasakian Metric

In this paper, for a Finsler manifold (M, F) with a Finsler metric gij(x, y) we shall consider a generalized Lagrange metrics (FGL-metrics) as the form *gij(x, y) = gij(x, y) + σ(x, y)Bi(x, y)Bj(x, y) on TM. Then we shall consider a Riemannian manifold (TM, *G) in which *G is a generalized Sasakian metric of *g on [Formula: see text]. Then we restrict the above FGL-metrics to a submanifold of [Formula: see text], and show that it admits a GL-metric structure. Then we shall find a necessary and sufficient condition for this submanifold to be totally geodesic.

On 4-th root metrics of isotropic scalar curvature

2020 ◽  
Vol 70 (1) ◽  
pp. 161-172
Author(s):  
Akbar Tayebi
Keyword(s):  
Scalar Curvature ◽  
Flag Curvature ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Conformal Change ◽  
Necessary And Sufficient

AbstractIn this paper, we prove that every non-Riemannian 4-th root metric of isotropic scalar curvature has vanishing scalar curvature. Then, we show that every 4-th root metric of weakly isotropic flag curvature has vanishing scalar curvature. Finally, we find the necessary and sufficient condition under which the conformal change of a 4-th root metric is of isotropic scalar curvature.

scholarly journals On the ME-manifold inn-*g-UFT and its conformal change

1994 ◽  
Vol 17 (1) ◽  
pp. 79-90
Author(s):  
Kyung Tae Chung ◽  
Gwang Sik Eun
Keyword(s):  
Riemannian Manifold ◽  
Geometric Structure ◽  
Tensor Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Unique Existence ◽  
Conformal Change ◽  
Necessary And Sufficient

An Einstein's connection which takes the form (3.1) is called an ME-connection. A generalizedn-dimensional Riemannian manifoldXnon which the differential geometric structure is imposed by a tensor field*gλνthrough a unique ME-connection subject to the conditions of Agreement (4.1) is called*g-ME-manifold and we denote it by*g-MEXn. The purpose of the present paper is to introduce this new concept of*g-MEXnand investigate its properties. In this paper, we first prove a necessary and sufficient condition for the unique existence of ME-connection inXn, and derive a surveyable tensorial representation of the ME-connection. In the second, we investigate the conformal change of*g-MEXnand present a useful tensorial representation of the conformal change of the ME-connection.

On generalized 4-th root metrics of isotropic scalar curvature

2018 ◽  
Vol 68 (4) ◽  
pp. 907-928 ◽  
Author(s):  
Akbar Tayebi
Keyword(s):  
Scalar Curvature ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Sufficient Condition ◽  
Riemannian Curvature ◽  
Necessary And Sufficient Condition ◽  
Conformal Change ◽  
Riemannian Curvature Tensor ◽  
Necessary And Sufficient

AbstractBy an interesting physical perspective and a suitable contraction of the Riemannian curvature tensor in Finsler geometry, Akbar-Zadeh introduced the notion of scalar curvature for the Finsler metrics. A Finsler metric is called of isotropic scalar curvature if the scalar curvature depends on the position only. In this paper, we study the class of generalized 4-th root metrics. These metrics generalize 4-th root metrics which are used in Biology as ecological metrics. We find the necessary and sufficient condition under which a generalized 4-th root metric is of isotropic scalar curvature. Then, we find the necessary and sufficient condition under which the conformal change of a generalized 4-th root metric is of isotropic scalar curvature. Finally, we characterize the Bryant metrics of isotropic scalar curvature.

Almost pseudo-Q-symmetric semi-Riemannian manifolds

2018 ◽  
Vol 15 (07) ◽  
pp. 1850117
Author(s):  
Ljubica Velimirović ◽  
Pradip Majhi ◽  
Uday Chand De
Keyword(s):  
Perfect Fluid ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Einstein Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Properties ◽  
Dust Fluid ◽  
Necessary And Sufficient ◽  
Fluid Spacetimes

The object of the present paper is to study almost pseudo-[Formula: see text]-symmetric manifolds [Formula: see text]. Some geometric properties have been studied which recover some known results of pseudo [Formula: see text]-symmetric manifolds. We obtain a necessary and sufficient condition for the [Formula: see text]-curvature tensor to be recurrent in [Formula: see text]. Also, we provide several interesting results. Among others, we prove that a Ricci symmetric [Formula: see text] is an Einstein manifold under certain condition. Moreover we deal with [Formula: see text]-flat perfect fluid, dust fluid and radiation era perfect fluid spacetimes respectively. As a consequence, we obtain some important results. Finally, we consider [Formula: see text]-spacetimes.

scholarly journals Remarks on metallic warped product manifolds

2018 ◽  
Vol 33 (2) ◽  
pp. 269
Author(s):  
Adara-Monica Blaga ◽  
Cristina-Elena Hretcanu
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Metallic Structure ◽  
Necessary And Sufficient

We characterize the metallic structure on the product of two metallic manifolds in terms of metallic maps and provide a necessary and sufficient condition for the warped product of two locally metallic Riemannian manifolds to be locally metallic. The particular case of product manifolds is discussed and an example of metallic warped product Riemannian manifold is provided.

scholarly journals The Identification of Convex Function on Riemannian Manifold

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Li Zou ◽  
Xin Wen ◽  
Hamid Reza Karimi ◽  
Yan Shi
Keyword(s):  
Riemannian Manifold ◽  
Convex Function ◽  
Global Minimum ◽  
Directional Derivative ◽  
Inequality Constraints ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Gradient ◽  
Necessary And Sufficient ◽  
Global Minimum Point

The necessary and sufficient condition of convex function is significant in nonlinear convex programming. This paper presents the identification of convex function on Riemannian manifold by use of Penot generalized directional derivative and the Clarke generalized gradient. This paper also presents a method for judging whether a point is the global minimum point in the inequality constraints. Our objective here is to extend the content and proof the necessary and sufficient condition of convex function to Riemannian manifolds.

scholarly journals Two theorems on(ϵ)-Sasakian manifolds

1998 ◽  
Vol 21 (2) ◽  
pp. 249-254 ◽  
Author(s):  
Xu Xufeng ◽  
Chao Xiaoli
Keyword(s):  
Riemannian Manifold ◽  
Sasakian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Sasakian Manifolds ◽  
Necessary And Sufficient ◽  
Kaehlerian Manifold

In this paper, We prove that every(ϵ)-sasakian manifold is a hypersurface of an indefinite kaehlerian manifold, and give a necessary and sufficient condition for a Riemannian manifold to be an(ϵ)-sasakian manifold.

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