A Note on Randers Metrics of Scalar Flag Curvature

2012 ◽  
Vol 55 (3) ◽  
pp. 474-486 ◽  
Author(s):  
Bin Chen ◽  
Lili Zhao
Keyword(s):  
Scalar Curvature ◽  
Three Dimensional ◽  
Constant Scalar Curvature ◽  
Flag Curvature ◽  
Projectively Flat ◽  
Randers Metrics

AbstractSome families of Randers metrics of scalar flag curvature are studied in this paper. Explicit examples that are neither locally projectively flat nor of isotropic S-curvature are given. Certain Randers metrics with Einstein α are considered and proved to be complex. Three dimensional Randers manifolds, with α having constant scalar curvature, are studied.

scholarly journals ON THE RANDERS METRICS ON TWO-STEP HOMOGENEOUS NILMANIFOLDS OF DIMENSION FIVE

2011 ◽  
Vol 08 (03) ◽  
pp. 501-510 ◽  
Author(s):  
HAMID REZA SALIMI MOGHADDAM
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Lie Groups ◽  
Three Dimensional ◽  
Flag Curvature ◽  
Nilpotent Lie Groups ◽  
Randers Metric ◽  
Levi Civita Connection ◽  
Simply Connected ◽  
Randers Metrics

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.

scholarly journals RANDERS METRICS OF SCALAR FLAG CURVATURE

2009 ◽  
Vol 87 (3) ◽  
pp. 359-370 ◽  
Author(s):  
XINYUE CHENG ◽  
ZHONGMIN SHEN
Keyword(s):  
Flag Curvature ◽  
Finsler Metrics ◽  
Important Class ◽  
Constant Flag Curvature ◽  
Projectively Flat ◽  
Randers Metrics

AbstractWe study an important class of Finsler metrics, namely, Randers metrics. We classify Randers metrics of scalar flag curvature whose S-curvatures are isotropic. This class of Randers metrics contains all projectively flat Randers metrics with isotropic S-curvature and Randers metrics of constant flag curvature.

Yamabe solitons and gradient Yamabe solitons on three-dimensional N(k)-contact manifolds

2020 ◽  
Vol 17 (12) ◽  
pp. 2050177
Author(s):  
Young Jin Suh ◽  
Uday Chand De
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Three Dimensional ◽  
Constant Scalar Curvature ◽  
Infinitesimal Transformation ◽  
Contact Metric Manifold ◽  
Contact Manifolds ◽  
Flow Vector ◽  
Yamabe Solitons ◽  
Yamabe Soliton

If a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] admits a Yamabe soliton of type [Formula: see text], then the manifold has a constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, either [Formula: see text] has a constant curvature [Formula: see text] or the flow vector field [Formula: see text] is a strict contact infinitesimal transformation. Also, we prove that if the metric of a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] admits a gradient Yamabe soliton, then either the manifold is flat or the scalar curvature is constant. Moreover, either the potential function is constant or the manifold is of constant sectional curvature [Formula: see text]. Finally, we have given an example to verify our result.

The Projectively Flat Conditions of One Special Class (α, β)-Metrics

2013 ◽  
Vol 756-759 ◽  
pp. 2528-2532
Author(s):  
Wen Jing Zhao ◽  
Yan Yan ◽  
Li Nan Shi ◽  
Bo Chao Qu
Keyword(s):  
Special Class ◽  
Riemannian Metric ◽  
Class I ◽  
Flag Curvature ◽  
Finsler Metrics ◽  
Important Class ◽  
Randers Metric ◽  
New Class ◽  
Projectively Flat ◽  
Randers Metrics

The-metric is an important class of Finsler metrics including Randers metric as the simplest class, and many people research the Randers metrics. In this paper, we study a new class of Finsler metrics in the form ,Whereis a Riemannian metric, is a 1-form. Bengling Li had introduced the projective flat of the-Metric F. We find another method which is about flag curvature to prove the projective flat conditions of this kind of-metric.

CONFORMAL TRANSFORMATIONS OF COMPACT SELF-DUAL MANIFOLDS

1994 ◽  
Vol 05 (01) ◽  
pp. 125-140 ◽  
Author(s):  
Y. S. POON
Keyword(s):  
Scalar Curvature ◽  
Symmetry Group ◽  
Positive Constant ◽  
Three Dimensional ◽  
Constant Scalar Curvature ◽  
Topological Classification ◽  
Conformal Transformations ◽  
Conformal Class ◽  
Zero Scalar Curvature

We prove that when the dimension of the group of conformal transformations of a compact self-dual manifold is at least three, the conformal class contains either a metric with positive constant scalar curvature or a metric with zero scalar curvature. This result is combined with a topological classification of 4-manifolds to provide a complete geometrical classification of the compact self-dual manifolds whose symmetry group is at least three-dimensional.

An Einstein-like metric on almost Кenmotsu manifolds

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4695-4702 ◽  
Author(s):  
Yaning Wang ◽  
Wenjie Wang
Keyword(s):  
Scalar Curvature ◽  
Hyperbolic Space ◽  
Critical Condition ◽  
Three Dimensional ◽  
Constant Scalar Curvature ◽  
Kenmotsu Manifold ◽  
Almost Kenmotsu Manifold

In this paper, we prove that if the metric of a three-dimensional (k,?)'-almost Kenmotsu manifold satisfies the Miao-Tam critical condition, then the manifold is locally isometric to the hyperbolic space H3(-1). Moreover, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation satisfies the Miao-Tam critical condition, then the manifold is either of constant scalar curvature or Einstein. Some corollaries of main results are also given.

Projectively flat Randers metrics with constant flag curvature

2003 ◽  
Vol 325 (1) ◽  
pp. 19-30 ◽  
Author(s):  
Zhongmin Shen
Keyword(s):  
Flag Curvature ◽  
Constant Flag Curvature ◽  
Projectively Flat ◽  
Randers Metrics

Randers Metrics of Constant Scalar Curvature

2013 ◽  
Vol 56 (3) ◽  
pp. 615-620
Author(s):  
Sevim Esra Sengelen ◽  
Zhongmin Shen
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Special Class ◽  
Constant Scalar Curvature ◽  
Riemannian Metric ◽  
Finsler Metrics ◽  
Randers Metric ◽  
Randers Metrics

Abstract.Randers metrics are a special class of Finsler metrics. Every Randers metric can be expressed in terms of a Riemannian metric and a vector field via Zermelo navigation. In this paper, we show that a Randers metric has constant scalar curvature if the Riemannian metric has constant scalar curvature and the vector field is homothetic

RANDERS METRICS OF REVERSIBLE CURVATURE

2013 ◽  
Vol 24 (01) ◽  
pp. 1350006 ◽  
Author(s):  
ZHONGMIN SHEN ◽  
GUOJUN YANG
Keyword(s):  
Local Structure ◽  
Flag Curvature ◽  
3 Dimensional ◽  
Randers Metric ◽  
Projectively Flat ◽  
Randers Metrics

In this paper, we introduce the notions of R-reversibility and Ricci-reversibility. We prove that Randers metrics are R-reversible or Ricci-reversible if and only if they are R-quadratic or Ricci-quadratic, respectively. Besides, we discuss the properties of Ricci- or R-reversible Randers metrics which are also weakly Einsteinian, or Douglassian, or of scalar flag curvature. In particular, we determine the local structure of Randers metrics which are Ricci-reversible and locally projectively flat, and prove that an n (≥ 3)-dimensional Ricci-reversible Randers metric of non-zero scalar flag curvature is locally projectively flat.

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

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