Isotropic S-curvature Research Articles

On a class of two-dimensional Einstein Finsler metrics of vanishing S-curvature

An (α,β)-metric is defined by a Riemannian metric α and 1-form β. We have characterized a class of two-dimensional (α,β)-metrics of isotropic (thus vanishing) S-curvature. In this paper, we determine the local structure of those metrics and show that those metrics are Einsteinian (equivalently, of isotropic flag curvature) but not Ricci-flat in general.

Four families of projectively flat Finsler metrics with $$\mathbf{K}=1$$ K = 1 and their non-Riemannian curvature properties

For the first time, Bryant introduced a locally projectively flat non-Riemannian Finsler metric with flag curvature $$\mathbf{K}=1$$ on $${\mathbb {S}}^2$$ . In this paper, we construct four new non-Riemannian families of projectively flat Finsler metrics with flag curvature $$\mathbf{K}=1$$ . These metrics belong to the class of generalized 4-th root metrics. The class of generalized 4-th root metrics is an extension of the class of 4-th root metrics which has close relations with General Relativity and Seismic Ray Theory. In this class, we characterize Finsler metrics of isotropic flag curvature. We find the necessary and sufficient condition under which a metric in this class be weakly Einstein. In this class, we characterize Finsler metrics of isotropic S-curvature and almost vanishing $$\Xi $$ -curvature. Finally, we find the necessary and sufficient condition under which a Finsler metric in this class has vanishing projective Ricci curvature.

On a class of two-dimensional Finsler manifolds of isotropic S-curvature

For an (α, β)-metric (non-Randers type) of isotropic S-curvature on an n-dimensional manifold with non-constant norm ║β║ α , we first show that n = 2, and then we characterize such a class of two-dimensional (α, β)-manifolds with some PDEs, and also construct some examples for such a class.

On a class of spherically symmetric Finsler metrics with isotropic S-curvature

S-curvature is an important non-Riemannian quantity in Finsler geometry. In this paper, we classify a class of spherically symmetric Finsler metrics with isotropic S-curvature.

On curvatures of homogeneous Finsler-Kropina space

In Finsler geometry, one of the central problem is to study curvature properties of special class of Finsler spaces such as homogeneous-Finsler spaces. In this paper, we give an explicit formula of the S-curvature of homogeneous Finsler space with Kropina metric and proved that a homogeneous Kropina space with almost isotropic S-curvature must have vanishing S-curvature. We also derived an explicit formula of the mean Berwald curvature Eij of homogeneous Finsler-Kropina space.

Properties of the Riemannian Curvature of (α, β)-Metrics

In this paper, we discuss some important properties of the Riemannian curvature of (α, β)-metrics. When the dimension of the manifold is greater than 2, we classify Randers metrics of weakly isotropic flag curvature (that is, Randers metrics of scalar flag curvature with isotropic S-curvature). Further, we characterize (α, β)-metrics of scalar flag curvature with isotropic S-curvature. We also characterize Einstein (α, β)-metrics and determine completely the local structure of Ricci-flat Douglas (α, β)-metrics when the dimension dim M ≥ 3.

On Kropina metrics with non-Riemannian curvature properties

In this paper, we first study two significant non-Riemannian quantities Ξ-curvature and H-curvature and show that a Kropina metric is of almost vanishing Ξ-curvature or H-curvature if and only if it is of isotropic S-curvature. Further, we prove that a Kropina metric F is a Douglas metric if and only if the conformally related metric F˜=eκ(x)F is also Douglas metric. Finally, we classify the Kropina metrics with related isotropic weak Landsberg curvature.

The spherically symmetric Finsler metrics with isotropic S-curvature

In this paper, we investigate the spherically symmetric Finsler metrics with isotropic S-curvature and obtain a characterized equation. As an application, we prove that these metrics with Douglas type must be Randers metrics or Berwald metrics. This result leads to two classification theorems.

KILLING FRAMES AND S-CURVATURE OF HOMOGENEOUS FINSLER SPACES

AbstractIn this paper, we first deduce a formula of S-curvature of homogeneous Finsler spaces in terms of Killing vector fields. Then we prove that a homogeneous Finsler space has isotropic S-curvature if and only if it has vanishing S-curvature. In the special case that the homogeneous Finsler space is a Randers space, we give an explicit formula which coincides with the previous formula obtained by the second author using other methods.

ON THE SECOND APPROXIMATE MATSUMOTO METRIC

In this paper, we study the second approximate Matsumoto metric F = <TEX>${\alpha}+{\beta}+{\beta}^2/{\alpha}+{\beta}^3/{\alpha}^2$</TEX> on a manifold M. We prove that F is of scalar flag curvature and isotropic S-curvature if and only if it is isotropic Berwald metric with almost isotropic flag curvature.

Special projective Lichnérowicz–Obata theorem for Randers spaces

It is proved that either every special projective vector field V on a Randers space (M,F=α+β) is a conformal vector field of the Riemannian metric α2−β2, or F is of isotropic S-curvature. This result is applied to establish a projective Lichnérowicz–Obata-type result on the closed manifolds with generic Randers metrics.

ON THE PROJECTIVE ALGEBRA OF SOME (α, β)-METRICS OF ISOTROPIC S-CURVATURE

In this paper, we study projective algebra, p(M, F), of special (α, β)-metrics. The projective algebra of a Finsler space is a finite-dimensional Lie algebra with respect to the usual Lie bracket. We characterize p(M, F) of Matsumoto and square metrics of isotropic S-curvature of dimension n ≥ 3 as a certain Lie sub-algebra of the Killing algebra k(M, α). We also show that F has a maximum projective symmetry if and only if F either is a Riemannian metric of constant sectional curvature or locally Minkowskian.

ON THE RIEMANN CURVATURE OPERATORS IN RANDERS SPACES

The Riemann curvature in Riemann–Finsler geometry can be regarded as a collection of linear operators on the tangent spaces. The algebraic properties of these operators may be linked to the geometry and the topology of the underlying space. The principal curvatures of a Finsler space (M, F) at a point x are the eigenvalues of the Riemann curvature operator at x. They are real functions κ on the slit tangent manifold TM0. A principal curvature κ(x, y) is said to be isotropic (respectively, quadratic) if κ(x, y)/F(x, y) is a function of x only (respectively, κ(x, y) is quadratic with respect to y). On the other hand, the Randers metrics are the most popular and prominent metrics in pure and applied disciplines. Here, it is proved that if a Randers metric admits an isotropic principal curvature, then F is of isotropic S-curvature. The same result is also established for F to admit a quadratic principal curvature. These results extend Shen's verbal results about Randers metrics of scalar flag curvature K = K(x) as well as those Randers metrics with quadratic Riemann curvature operator. The Riemann curvature [Formula: see text] may be broken into two operators [Formula: see text] and [Formula: see text]. The isotropic and quadratic principal curvature are characterized in terms of the eigenvalues of [Formula: see text] and [Formula: see text].

On conformal transformations between two [formula omitted]-metrics

In this paper, we characterize the conformal transformations between two (α,β)-metrics. Suppose that F is an (α,β)-metric of non-Randers type and is conformally related to F˜, that is, F˜=eκ(x)F, where κ:=κ(x) is a scalar function on the manifold. We prove that, if F is a Douglas metric, then F˜ is also a Douglas metric if and only if the conformal transformation is a homothety. Further, we prove that, if F is of isotropic S-curvature, then F˜ is also of isotropic S-curvature if and only if the conformal transformation is a homothety.

Characterization of locally dually flat Matsumoto metric

The concept of locally dually flat Finsler metrics are originated from information geometry. As we know, α, β -metrics defined by a Riemannian metric α and 1-form β form an important class of Finsler metrics. In this paper, we study and characterize Locally dually flat Matsumoto metric F = α/α − β with isotropic S-curvature which is not Riemannian.

AN IMPORTANT CLASS OF CONFORMALLY FLAT WEAK EINSTEIN FINSLER METRICS

In this paper, we study conformally flat (α, β)-metrics in the form F = αϕ(β/α), where α is a Riemannian metric and β is a 1-form on a C∞manifold M. We prove that if ϕ = ϕ(s) is a polynomial in s, the conformally flat weak Einstein (α, β)-metric must be either a locally Minkowski metric or a Riemannian metric. Moreover, we prove that conformally flat (α, β)-metrics with isotropic S-curvature are also either locally Minkowski metrics or Riemannian metrics.

Measurable [formula omitted]-spaces with vanishing S-curvature

In this paper, we give a necessary and sufficient condition that an (α,β)-space admits a measure μ with vanishing S-curvature everywhere. It is shown that the measure of such an (α,β)-space must coincide with the Riemannian volume measure of α up to a constant multiplication. Moreover, the characterization of (α,β)-spaces admitting a measure μ with weak isotropic S-curvature is given.

On locally dually flat (α, β)-metrics with isotropic S-curvature

In this paper, we consider locally dually flat (α, β)-metrics with isotropic S-curvature and find some necessary and sufficient conditions under which these metrics reduce to locally Minkowskian metrics.

A Note on Randers Metrics of Scalar Flag Curvature

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.

共形平坦的Randers度量

In this paper, the properties of conformally flat Randers metrics are studied. It is proved that any conformally flat Randers metric of scalar curvature is projectively flat. Moreover, the complete classification of such Randers metrics is given. It is also shown that there is no non-trivial conformally flat Randers metrics of almost isotropic S-curvature.