Finsler metrics with special Riemannian curvature properties

The Weyl curvature is one of the fundamental quantities in Finsler geometry because it is a projective invariant. By determining the Weyl curvature of a class of Finsler metrics, we find a lot of Finsler metrics of quadratic Weyl curvature which are non-trivial in the sense that they are not of quadratic Riemann curvature.     

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.     

Examples of Finsler metrics with special curvature properties

We give a lot of new Finsler metrics of quadratic Weyl curvature which are non‐trivial in the sense that these metrics are not of Weyl type. We also manufacture new Weyl metrics.     

Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties

In this paper, we study Finsler metrics of scalar flag curvature. We find that a non-Riemannian quantity is closely related to the flag curvature. We show that the flag curvature is weakly isotropic if and only if this non-Riemannian quantity takes a special form. This will lead to a better understanding on Finsler metrics of scalar flag curvature.      

The explicit construction of Finsler metrics with special curvature properties

The S-curvature is one of most important non-Riemannian quantities in Finsler geometry. It delicately related to Riemannian quantities. This note gives an explicit construction of 3-parameter family of non-locally projectively flat Finsler metrics of non-constant isotropic S-curvature. The necessary and sufficient condition that these Finsler metrics are of constant flag curvature is given.     

On doubly warped product Finsler manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold.     

Two-dimensional Finsler metrics with constant flag curvature

 We construct infinitely many two-dimensional Finsler metrics on 𝕊² and 𝔻² with non-zero constant flag curvature. They are all not locally projectively flat. Received: 10 October 2001 / Revised version: 15 July 2002     

Finsler metrics with positive constant flag curvature

We showed that any reversible Finsler metric with positive constant flag curvature must be Riemannian. Received: 18 August 2008     

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.     

Inverse curvature flows in Riemannian warped products

The long-time existence and umbilicity estimates for compact, graphical solutions to expanding curvature flows are deduced in Riemannian warped products of a real interval with a compact fibre. Notably we do not assume the ambient manifold to be rotationally symmetric, nor the radial curvature to converge, nor a lower bound on the ambient sectional curvature. The inverse speeds are given by powers $0<p\le 1$ of a curvature function satisfying few common properties.     

一些球对称射影平坦的Finsler度量的构造

研究刻画球对称Finsler度量的射影平坦性质的偏微分方程,通过对射影平坦Finsler度量PDE的研究,构造了两类球对称射影平坦Finsler度量,得到了一些球对称的射影平坦Finsler度量,并进一步给出这些Finsler度量的射影因子和旗曲率.     

Smoothing Riemannian metrics with Ricci curvature bounds

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds. The authors were supported in part by NSF Grant. The first author was also supported in part by Alfred P. Sloan Fellowship This article was processed by the author using the LATEX style filecljourl from Springer-Verlag.     

Reversible Finsler metrics of constant flag curvature

Without the restriction of quadratic form as a Riemannian metric, a Finsler metric $F=F(x,y)$ on a smooth manifold M can be reversible (symmetric in y) or not. Reversible Finsler metrics have different properties from Riemannian metrics though it seems they are very close to Riemannian metrics. Hilbert metric is the famous reversible Finsler metric of negative constant flag curvature in the history, and it is projectively flat. Then it is natural to ask the question how to classify reversible projectively flat Finsler metrics of constant flag curvature and give more new examples? In this paper, we answer the above question by giving the classification when the flag curvature $\mathbf{K}=-1,0,1$ respectively. Especially, for the case when $\mathbf{K}=-1$, we show that the only reversible projectively flat Finsler metrics are just Hilbert metrics. For the case when $\mathbf{K}=1$, we give an algebraic way to construct explicit metric function by solving algebraic equations, such as by solving a quartic equation. When the flag curvature is zero, it is much easier to construct reversible projectively flat Finsler metrics than before.     

On spherically symmetric Finsler metrics with vanishing Douglas curvature

We obtain the differential equation that characterizes the spherically symmetric Finsler metrics with vanishing Douglas curvature. By solving this equation, we obtain all the spherically symmetric Douglas metrics. Many explicit examples are included.     

Some remarks on the Riemannian metrics on warped products

The purpose of the present paper is to study the following problem. Given a warped product M:=B×_f F, F=ø, give conditions that B and f must satisfy for the existence of the Riemannian simple double of M.This research was supported in part by Non-Directed Research Fund, Korea Research Foundation, 1989.     

Warped products with special Riemannian curvature

We study the geometry of particular classes of Riemannian manifolds obtained as warped products. We focus on the case of constant curvature which is completely classified and on the Einstein case. This study provides nontrivial instances of Einstein manifolds which are warped product of Einstein factors.Supported by a grant from Università di Parma     

Finsler metrics with constant (or scalar) flag curvature

By finding Killing vector fields of general Bryant??s metric we give a lot of new Finsler metrics of constant (or scalar) flag curvature and determine their scalar curvature.     

Finsler metrics with constant （or scalar） flag curvature

A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K =3c + σ where σ and c are scalar functions on M.In this paper,we establish the intrinsic re...     

Projectively flat Finsler metrics of constant flag curvature

Finsler metrics on an open subset in with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.