arXiv research

New Finsler metrics with constant flag curvature defined using Weyl-type curvature tensor.

Study Finsler metrics with Killing fields on constant flag curvature surfaces.

The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in R^n with negative flag curvature and constant S-curvature. In th…

Classifies minimal immersions from $S^2$ into specific flag manifolds.

Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.

The local structure of Finsler metrics of constant flag curvature have been historically mysterious. It is proved that every Matsumoto metric of constant flag curvature on a manifold of dimension n \geq 3 is either Riemannian or locally Minkowskian.

Study geodesics on spheres with constant curvature, showing integrability and invariant properties.

A new axiom for Finsler geometry leads to constant flag curvature.

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 are found to have constant negative flag curvature.

Study spherically symmetric Finsler metrics with specific curvature properties.

Paper proves algebraic condition for Finsler metrics with constant curvature.

Study on constant curvature immersions of surfaces into flag manifolds.

Paper finds conditions for generalized Kropina spaces to have constant curvature.

In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary dif…

In this paper we study the flag curvature of a particular class of Finsler metrics called general $(α,β)$-metrics, which are defined by a Riemannian metric $α$ and a $1$-form $β$. The classification of such metrics with constant flag curvature are completely determined under some suitable conditions, which make them be…

This paper solves Hilbert's fourth problem for constant curvature metrics.

By using the Hawking Taub-NUT metric, this note gives an explicit construction of a 3-parameter family of Einstein Finsler metrics of non-constant flag curvature in terms of navigation representation.

In this paper by using left invariant Riemannian metrics on some 3-dimensional Lie groups we construct some complete non-Riemannian Berwald spaces of non-positive flag curvature and several families of geodesically complete locally Minkowskian spaces of zero constant flag curvature.

Recently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group can not be a compact Lie group, if the Finsler manifold of dimension $> 2$ has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holon…

Paper examines conditions for singular square metrics to have constant curvature.

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we classify a class of singular $(α,β)$-metrics which are locally projectively flat with constant flag curvature in dimension $n= 2$ and $n \ge 3$ respectively. Further, we determine t…

New Finslerian Ressiner-Nordstrom spacetime with constant flag curvature.

The paper finds and analyzes the Funk-Finsler structure in constant curvature spaces.

Study geodesics on positively curved Zoll surfaces.

Homogeneous Finsler spheres with constant curvature have specific geodesic properties.

The paper extends geodesic orbit sphere classification to Finsler geometry.

In this paper conditions for a Kropina structure to be of constant flag curvature are obtained.

The paper studies invariant functions and their relation to Landsberg surfaces.

This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. <p> The second remark is that there is a natural way to construct a (…

Study isoparametric hypersurfaces in a Randers sphere with constant flag curvature.

Wind Riemannian structures generalize Randers metrics and are classified for constant flag curvature.

In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Ran…

A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodes…

In this paper, we study locally projectively flat Finsler metrics with constant flag curvature ${\bf K}$. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when ${\bf K}=0$, ${\bf K}=-1$ and ${\bf K} =1$ are given respectively in an algebraic way.…

We say that a nonnegatively curved manifold $(M,g)$ has quarter pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem of Brendle and Schoe…

Finsler metrics of constant curvature characterized, with a Finslerian Beltrami Theorem.

The collection of all projective vector fields on a Finsler space $(M, F)$ is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by $p(M,F)$ and is the Lie algebra of the projective group $P(M,F)$. The projective algebra $p(M,F=α+β)$ of a Randers space is chara…

The abstract proves properties of Berwald spaces with non-zero flag curvature.

The study examines geodesic orbit Finsler spaces with non-negative flag curvature and (FP) condition, proving they are compact.

Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.

Complete Finsler spaces with negative Ricci curvature are reversible.

In this paper, we classify the spherically symmetric Berwald metrics in $\mathbb{R}^n$. For the spherically symmetric Landsberg metrics, we prove that there do not exist any non-Berwald metrics among the regular case. The partial differential equation systems which can respectively characterize the spherically symmetri…

It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, chara…

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 in…

Defines flag structures on real 3-manifolds and proves null curvature models.

Here, an extension of the Obata-Tanno's theorem to Finsler geometry is established and the following rigidity result is obtained; Every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to an $n$-sphere equipped with a certain Finsler metric, and vise versa.

Study projective symmetries in Finsler spaces, showing reductions and constant flag curvature.

In this paper, firstly we study some left invariant Riemannian metrics on para-hypercomplex 4-dimensional Lie groups. In each Lie group, the Levi-Civita connection and sectional curvature have been given explicitly. We also show these spaces have constant negative scalar curvatures. Then by using left invariant Riemann…