Isotropic S-curvature Research Articles

ON PROJECTIVELY FLAT GENERALIZED BERWALD (α, β)-METRICS

Every Berwald metric is a special generalized Berwald metric. In this paper, we study the class of projectively flat generalized Berwald (α; β)-metrics of isotropic S-curvature. We find some conditions under which this class of Finsler metrics reduces to the class of Berwald metrics.

On harmonic and asymptotically harmonic Finsler manifolds

In the present paper we introduce and investigate some types of harmonic Finsler manifolds and find out the interrelation between them. We give some characterizations of such spaces in terms of the Finsler mean curvature of geodesic spheres and the Finsler Laplacian of the distance function induced by the Finsler metric. We investigate some properties of the Finsler mean curvature of geodesic spheres of different radii. We prove that certain harmonic Finsler manifolds are of isotropic S-curvature and isotropic Ricci curvature. Moreover, we give some examples of non-Riemannian Finsler harmonic manifolds of constant flag curvature and constant S-curvature. Finally, we provide a technique to construct harmonic Finsler manifolds of Randers type.

On conformal transformations preserving the Ricci tensor in Finsler geometry

Here we obtain a classical integral formula on the conformal change of Finsler metrics. As an application, we obtain significant results depending on the sign of the Ricci scalars, for mean Landsberg surfaces and show there is no conformal transformation between two compact mean Landsberg surfaces, one of a non-positive Ricci scalar and another of a non-negative Ricci scalar, except for the case where both Ricci scalars are identically zero. Conformal transformations preserving the Ricci tensor are known as Liouville transformations. Here we show that a Liouville transformation between two compact mean Landsberg manifolds of isotropic S-curvature is homothetic. Moreover, every Liouville transformation between two compact Finsler n-manifolds of bounded mean value Cartan tensor is homothetic. These results are an extension of the results of M. Obata and S. T. Yau on Riemannian geometry and give a positive answer to a conjecture on Liouville's theorem.

On isoperimetric problem in 2-dimensional Randers space

In this paper, we prove that the circle centered at the origin in B2(δξ) is a proper maximum of the isoperimetric problem in a 2-dimensional Randers space endowed with 3-parameter family of non-locally projectively flat Finsler metrics of non-constant isotropic S-curvature.

Curvatures on Homogeneous Generalized Matsumoto Space

The curvature characteristics of particular classes of Finsler spaces, such as homogeneous Finsler spaces, are one of the major issues in Finsler geometry. In this paper, we have obtained the expression for S-curvature in homogeneous Finsler space with a generalized Matsumoto metric and demonstrated that the homogeneous generalized Matsumoto space with isotropic S-curvature has to vanish the S-curvature. We have also derived the expression for the mean Berwald curvature by using the formula of S-curvature.

On the Equivalence of Two Non-Riemannian Curvatures in Warped Product Finsler Metrics

In this paper we discuss the Busemann-Hausdorff volume form and Holmes-Thompson volume form for the warped product Finsler metrics. With the help of these volume forms we obtain the E-curvature and the S-curvature for this class of metrics. Further, we show that the notion of isotropic E-curvature and isotropic S-curvature are equivalent for this class of metrics.

On Almost Rational Finsler Metrics

We study a special class of Finsler metrics which we refer to as Almost Rational Finsler metrics (shortly, AR-Finsler metrics). We give necessary and sufficient conditions for an AR-Finsler manifold (M, F) to be Riemannian. The rationality of some Finsler geometric objects such as Cartan torsion, geodesic spray, Landsberg curvature and S-curvature is investigated. For a particular subfamily of AR-Finsler metrics we have proved that if F has isotropic S-curvature, then the S-curvature vanishes identically; if F has isotropic mean Landsberg curvature, then it is weakly Landsberg; if F is an Einstein metric, then it is Ricci-flat. Moreover, there exists no Randers AR-Finsler metric. Finally, we provide some nontrivial examples of AR-Finsler metrics.

On a class of weakly weighted Einstein metrics

The notion of general weighted Ricci curvatures appears naturally in many problems. The N-Ricci curvature and the projective Ricci curvature are just two special ones with totally different geometric meanings. In this paper, we study general weighted Ricci curvatures. We find that Randers metrics of certain isotropic weighted Ricci curvature must have isotropic S-curvature. Then we classify them via their navigation expressions. We also find equations that characterize Randers metrics of almost isotropic weighted Ricci curvature.

On weakly stretch Randers metrics

In this paper, we study weakly stretch Randers metrics. We find that any weakly stretch Randers metrics must be of isotropic S-curvature. Further, we also classify the weakly stretch Randers metrics with scalar flag curvature.

Finsler warped product metrics with isotropic S-curvature

The S-curvature is one of most important non-Riemannian quantities in Finsler geometry. In this paper, we study Finsler warped product metrics with isotropic S-curvature. We find the equation that characterizes Finsler warped product metrics of isotropic S-curvature. Then we explicitly construct infinitely many new non-spherically symmetric such metrics.

On non-positively curved homogeneous Finsler metrics

In this paper, we prove two rigidity results for non-positively curved homogeneous Finsler metrics. Our first main result yields an extension of Hu-Deng's well-known result proven for the Randers metrics. Indeed, we prove that every connected homogeneous Finsler space with non-positive flag curvature and isotropic S-curvature is Riemannian or locally Minkowskian. We extend the Szabó's rigidity theorem for Berwald surfaces and show that homogeneous isotropic Berwald metrics with non-positive flag curvature are Riemannian or locally Minkowskian. Our second main result is to show that every homogeneous (α,β)-metric with non-positive flag curvature and almost isotropic S-curvature is Riemannian or locally Minkowskian.

S-curvature and E-curvature of invariant (α,β)-metric on homogeneous space

Let F=αϕ(s),s=βα, be an invariant (α,β)-metric on a reductive homogeneous manifold G/H of dimension n≥3 such that ϕ(s) comes from a real analytic function. In this paper, we first obtain the necessary and sufficient conditions for F to be a weakly Berwald metric, and then prove that F has weakly isotropic S-curvature if and only if F has vanishing S-curvature. Based these results, we prove that S=0 if and only if E=0.

Yamabe Problem for Kropina Metrics

In this paper, we obtain the formula of scalar curvature for Kropina metrics. Then we prove that a Kropina metric of isotropic scalar curvature must be of isotropic S-curvature. We also focus on Yamabe problem in Kropina metrics and give a negative answer to the Yamabe problem on Kropina metrics with isotropic S-curvature. It turns out that C-reducible metrics do not satisfy Yamabe problem.

Weighted projective Ricci curvature in Finsler geometry

AbstractIn this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.

S-curvature of Doubly Warped Product of Finsler Manifolds

Doubly warped product of Finsler manifolds is useful in theoretical physics, particularly in general relativity. In this paper, we study doubly warped product of Finsler manifolds with isotropic mean Berwald curvature or weak isotropic S-curvature.

English

We characterize generalized Douglas-Weyl Randers metrics in terms of their Zermelo navigation data. Then, we study the Randers metrics induced by some important classes of almost contact metrics. Furthermore, we construct a family of generalized Douglas-Weyl Randers metrics which are not R-quadratic. We show that the Randers metric induced by a Kenmotsu manifold is a Douglas metric which is not of isotropic S-curvature. We show that the Randers metric induced by a Kenmotsu or Sasakian manifold is not Einsteinian. By using D-homothetic deformation of a Kenmotsu or Sasakian manifold, we construct a family of generalized Douglas-Weyl Randers metrics and show that the Lie group of projective transformations does not act transitively on the set of generalized Douglas-Weyl Randers metrics.

S - Curvature of Homogeneous Finsler space with special (a, b)-metrics

In this paper, we derive the formula for S-curvature of homogeneous Finsler space with special (α, β)-metric F=α+(β2/α) and further proved that the almost isotropic S-curvature must vanishes S-curvature. Also we find the explicit formula of the mean Berwald curvature.

On Spherically Symmetric Landsberg Metrics

ABSTRACTIn this paper, we have studied spherically symmetric Landsberg metrics with isotropic E-curvature and isotropic S-curvature. In the first case we have shown that the metric reduces to a Berwald metric and therefore, it is of vanishing E-curvature. In the second case we have completely classified the spherically symmetric Landsberg metric with isotropic S-curvature.

On a class of stretch metrics in Finsler Geometry

The class of stretch metrics contains the class of Landsberg metrics and the class of R-quadratic metrics. In this paper, we show that a regular non-Randers type (alpha , beta )-metric with vanishing S-curvature is stretchian if and only if it is Berwaldian. Let F be an almost regular non-Randers type (alpha , beta )-metric. Suppose that F is not a Berwald metric. Then, we find a family of stretch (alpha , beta )-metrics which is not Landsbergian. By presenting an example, we show that the mentioned facts do not hold for the Randers-type metrics. It follows that every regular (alpha , beta )-metric with isotropic S-curvature is R-quadratic if and only if it is a Berwald metric.

On general (α,β)-metrics with isotropic S-curvature

The S-curvature is an important non-Riemannian quantity in Finsler geometry. 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 characterize all of general (α,β)-metrics with isotropic S-curvature.