Related papers: On a Class of Two-Dimensional Douglas and Projecti…
In this paper, the geometric meaning of (alpha,beta)-norms is made clear. On this basis, we introduce a new class of Finsler metrics called general (alpha,beta)-metrics, which are defined by a Riemannian metric and an 1-form. These metrics…
Square metrics $F=\frac{(\alpha+\beta)^2}{\alpha}$ are a special class of Finsler metrics. It is the rate kind of metric category to be of excellent geometrical properties. In this paper, we discuss the so-called singular square metrics…
The $\Gamma$-limit for a sequence of length functionals associated with a one parameter family of Riemannian manifolds is computed analytically. The Riemannian manifold is of `two-phase' type, that is, the metric coefficient takes values in…
In this paper, we study almost regular Landsberg general $(\alpha,\beta)$-metrics in Finsler geometry. The corresponding equivalent equations are given. By solving the equations, we give the classification of Landsberg general…
In this paper, I will show how to use $\beta$-deformations to deal with dual flatness of $(\alpha,\beta)$-metrics. It is a natural continuation of the research on dually flat Randers metrics(see arxiv:1209.1150). $\beta$-deformations is a…
In this paper, we study a class of Finsler metrics called general $(\alpha,\beta)$-metrics, which are defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$. We classify this class of Finsler metrics with isotropic Berwald curvature…
We have shown that the Beltrami Theorem in Riemannian geometry is still true for square metrics if the dimension $n\ge 3$, namely, an $n(\ge 3)$-dimensional square metric is locally projectively flat if and only if it is of scalar flag…
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 $(\alpha,\beta)$-metrics which are locally projectively flat with…
In this paper, we study a new class of Finsler metrics, F=\alpha\phi(b^2,s), s:=\beta/\alpha, defined by a Riemannian metric \alpha and 1-form \beta. It is called general (\alpha, \beta) metric. In this paper, we assume \phi be coefficient…
A new geometrical definition of naturally reductive Finsler manifold using geodeic graph is proposed, with a possible generalization. Based on a construction from a recent paper by the authors, Finsler metrics based on naturally reductive…
We consider a special class of Finsler metrics --- square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that an analogue of the Beltrami Theorem in Riemannian geometry is still true for square metrics…
In this paper we classify all simply connected five dimensional nilpotent Lie groups which admit $(\alpha,\beta)$-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During…
We describe the $(\alpha,\beta)$-metrics whose the $T$-tensor vanishes ($T$-condition) and the $(\alpha,\beta)$-metrics that satisfy the $\sigma T$-condition $\sigma_hT^h_{ijk}=0$, where $\sigma_h=\frac{\partial \sigma}{\partial x^h}$ and…
A Riemannian metric is termed a Hessian metric if in some coordinate system it can be locally represented as the Hessian quadratic form of some locally defined smooth potential function. Under very mild extra technical conditions, we first…
The class of generalized Berwald metrics contains the class of Berwald metrics. In this paper, we characterize two-dimensional generalized Berwald $(\alpha, \beta)$-metrics with vanishing S-curvature. Let $F=\alpha\phi(s)$,…
In this paper we construct a new class of harmonic and asymptotically harmonic Finsler manifolds of $({\alpha},\beta)$-type. This class is defined by a Riemannian metric ${\alpha}$ and a special 1-form $\beta$.
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…
In this paper, we introduce the notion of Einstein-reversibility for Finsler met- rics. We study a class of p-power Finsler metrics determined by a Riemann metric and 1-form which are of Einstein-reversibility. It shows that such a class of…
I classify the Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature and whose geodesics are the great circles. Modulo diffeomorphism, there is a 2-parameter family of such Finsler structures, only one of which is…
In this paper, we prove that a strongly convex complex Finsler metric $F$ on a domain $D\subset\mathbb{C}^n$ is projectively flat (resp. dually flat) if and only if $F$ comes from a strongly convex complex Minkowski metric.