Finsler metrics with special curvature properties

In this paper, we discuss the relationship between the flag curvature and some non-Riemannian quantities of Finsler metrics of scalar curvature. In particular, we characterize projectively flat Finsler metrics with isotropic S-curvature.     

On Einstein Matsumoto metrics

Einstein metrics are solutions to Einstein field equation in General Relativity containing the Ricci-flat metrics. Einstein Finsler metrics which represent a non-Riemannian stage for the extensions of metric gravity, provide an interesting source of geometric issues and the (α,β)-metric is an important class of Finsler metrics appearing iteratively in physical studies. It is proved that every n-dimensional (n≥3) Einstein Matsumoto metric is a Ricci-flat metric with vanishing S-curvature. The main result can be regarded as a second Schur type Lemma for Matsumoto metrics.     

A global classification result for Randers metrics of scalar curvature on closed manifolds

One important problem in Finsler geometry is that of classifying Finsler metrics of scalar curvature. By investigating the second-order differential equation for a class of Randers metrics with isotropic S$S$-curvature, we give a global classification of these metrics of scalar curvature, generalizing a theorem previously only known in the case of locally projectively flat Randers metrics.     

On the non-Riemannian quantity H in Finsler geometry

In this paper, we study a new non-Riemannian quantity H defined by the S-curvature. We find that the non-Riemannian quantity is closely related to S-curvature. We characterize Randers metrics of almost isotropic S-curvature if and only if they have almost vanishing H-curvature. Furthermore, the Randers metrics actually have zero S-curvature if and only if they have vanishing H-curvature.     

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.     

On an Important Non-Riemannian Quantity in Finsler Geometry

In this paper, we study a non-Riemannian quantity ${\bar{{\bf E}}}$ -curvature. We prove that if F is a projectively flat Finsler metric of nonzero flag curvature, then it is Riemannian if and only if ${{\bar{\bf E}}}$ -curvature vanishes. Further, we characterize the Einstein-Douglas metrics with vanishing ${{\bar{\bf E}}}$ -curvature.     

On m-th root metrics with special curvature properties

In this Note, we prove that every m-th root Finsler metric with isotropic Landsberg curvature reduces to a Landsberg metric. Then, we show that every m-th root metric with almost vanishing H-curvature has vanishing H-curvature.     

Einstein Finsler metrics and killing vector fields on Riemannian manifolds

We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S ³ with Ric = 2F ², Ric = 0 and Ric = -2F ², respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.     

On the non-Riemannian quantity H of a Finsler metric

One of fundamental problems in Finsler geometry is to establish some delicate equations between Riemannian invariants and non-Riemannian invariants. Inspired by results due to Akbar-Zadeh etc., this note establishes a new fundamental equation between non-Riemannian quantity H and Riemannian quantities on a Finsler manifold. As its application, we show that all R-quadratic Finsler metrics have vanishing non-Riemannian invariant H generalizing result previously only known in the case of Randers metric.     

The S-curvature of homogeneous Randers spaces

In this paper, we give an explicit formula of the S-curvature of homogeneous Randers spaces and prove that a homogeneous Randers space with almost isotropic S-curvature must have vanishing S-curvature. As an application, we obtain a classification of homogeneous Randers space with almost isotropic S-curvature in some special cases. Some examples are also given.     

On Conformally Flat(α,β)-metrics with Special Curvature Properties

In this paper, we study a significant non-Riemannian quantity Ξ-curvature, which is defined by S-curvature. Firstly, we obtain the formula of Ξ-curvature for(α, β)-metrics. Based on it, we show that the Ξ-curvature vanishes for a class of(α, β)-metrics. In the end, we get the relation ofΞ-curvature for conformally related Finsler metrics, and classify conformally flat(α, β)-metrics with almost vanishing Ξ-curvature.     

射影Ricci平坦的Kropina度量

本文研究和刻画了射影Ricci平坦的Kropina度量.利用Kropina度量的S-曲率和Ricci曲率的公式,得到了Kropina度量的射影Ricci曲率公式.在此基础上得到了Kropina度量是射影Ricci平坦度量的充分必要条件.进一步,作为自然的应用,本文研究和刻画了由一个黎曼度量和一个具有常数长度的Killing 1-形式定义的射影Ricci平坦的Kropina度量,也刻画了具有迷向S-曲率的射影Ricci平坦的Kropina度量.在这种情形下,Kropina度量是Ricci平坦度量.     

A new class of projectively flat Finsler metrics with constant flag curvature K=1

In this paper, we consider a class of Finsler metrics which obtained by Kropina change of the class of generalized m-th root Finsler metrics. We classify projectively flat Finsler metrics in this class of metrics. Then under a condition, we show that every projectively flat Finsler metric in this class with constant flag curvature is locally Minkowskian. Finally, we find necessary and sufficient condition under which this class of metrics be locally dually flat.     

Singular solutions for the constant Q-curvature problem

This paper is devoted to the construction of weak solutions to the singular constant Q-curvature problem. We build on several tools developed in the last years. This is the first construction of singular metrics on closed manifolds of sufficiently large dimension with constant (positive) Q-curvature.     

Projective spherically symmetric Finsler metrics with constant flag curvature in R n

We investigate projective spherically symmetric Finsler metrics with constant flag curvature in R ⁿ and give the complete classification theorems. Furthermore, a new class of Finsler metrics with two parameters on n-dimensional disk is found to have constant negative flag curvature.     

On conformal complex Finsler metrics

In this paper, we study conformal transformations in complex Finsler geometry. We first prove that two weakly Kähler Finsler metrics cannot be conformal. Moreover, we give a necessary and sufficient condition for a strongly pseudoconvex complex Finsler metric to be locally conformal weakly Kähler Finsler. Finally, we discuss conformal transformations of a strongly pseudoconvex complex Finsler metric, which preserve the geodesics, holomorphic S curvatures and mean Landsberg tensors.

     

The global structure of locally convex hypersurfaces in Finsler—Hadamard manifolds

Locally convex compact immersed hypersurfaces in the Finsler—Hadamard space with bounded T-curvature are considered. Under certain conditions on normal curvatures, such hypersurfaces are proved to be convex, embedded, and homeomorphic to the sphere. To this end, the Rauch theorem is generalized to exponential maps of hypersurfaces and the convexity of parallel hypersurfaces is proved.     

Uniqueness and non-existence theorems for conformally invariant equations

By using the equivalent integral form for the Q-curvature equation, we generalize the well-known non-existence results on two-dimensional Gaussian curvature equation to all dimensional Q-curvature equation. Somehow, we introduce a new approach to Q-curvature equation which is higher order and even pseudo-differential equation. As a by-product, we do classify the solutions for Q=1 solutions on Sⁿ as well as on Rⁿ with necessary growth rate assumption.     

Constant Q-curvature metrics near the hyperbolic metric

Let (M,g)$(M,g)$ be a Poincaré–Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant Q-curvature in the conformal class of an asymptotically hyperbolic metric close enough to g. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant Q-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.     

On projective symmetries on Finsler spaces

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of I-invariant projective vector fields. The sub-algebra of the C-projective vector fields, leaving the H-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature.