arXiv subjects

Shaoqiang Deng

Publications and source records attributed to Shaoqiang Deng.

31 records · Page 2Linked to original sources

arXiv2014

Normal homogeneous Finsler spaces

In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Finsler spaces of positive flag curvature to an algebraic problem. The main result of this paper is a classification of positively curved normal homogeneous Finsler spaces. It turns out that a coset space $G/H$ admits a positively curved normal homogeneous Finsler metric if and only if it admits a positively curved normal homogeneous Riemannian metric. We will also give a complete description of the coset spaces admitting non-Riemannian positively curved normal homogeneous Finsler spaces.

math.DG↗

arXiv2014

The Landsberg equation of a Finsler space

Given a Finsler space, we introduce a system of partial differential equations, called the Landsberg equation. Based on a careful analysis of the Landsberg equation and the observation that the solution space is invariant under the linear isometries of the tangent Minkowski spaces, we prove that an $(α_1, α_2)$-metric of the Landsberg type must be a Berwald metric. This shows that the hunting for a unicorn, one of the longest standing open problem in Finsler geometry, cannot be successful even in the very broad class of $(α_1,α_2)$-metrics.

math.DG↗

arXiv2014

$(α_1,α_2)$-Spaces and Clifford-Wolf Homogeneity

In this paper, we introduce a new type of Finsler metrics, called $(α_1,α_2)$-metrics. We define the notion of the good datum of a homogeneous $(α_1,α_2)$-metric and use that to study the geometric properties. In particular, we give a formula of the S-curvature and deduce a condition for the S-curvature to be vanishing identically. Moreover, we consider the restrictive Clifford-Wolf homogeneity of left invariant $(α_1,α_2)$-metrics on compact connected simple Lie groups. We prove that, in some special cases, a restrictively Clifford-Wolf homogeneous $(α_1,α_2)$-metric must be Riemannian. An unexpected interesting observation contained in the proof reveals the fact that the S-curvature may play an important role in the study of Clifford-Wolf homogeneity in Finsler geometry.

math.DG↗

arXiv2013

Clifford-Wolf homogeneous left invariant $(α,β)$-metrics on compact semi-simple Lie groups

Let $(M,F)$ be a connected Finsler space. An isometry of $(M,F)$ is called a Clifford-Wolf translation (or simply CW-translation) if it moves all points the same distance. The compact Finsler space $(M,F)$ is called restrictively Clifford-Wolf homogeneous (restrictively CW-homogeneous) if for any two sufficiently close points $x_1,x_2\in M$, there exists a CW-translation $σ$ such that $σ(x_1)=x_2$. In this paper, we define the good normalized datum for a homogeneous non-Riemannian $(α,β)$-space, and use it to study the restrictive CW-homogeneity of left invariant $(α,β)$-metrics on a compact connected semisimple Lie group. We prove that a left invariant restrictively CW-homogeneous $(α,β)$-metric on a compact semisimple Lie group must be of the Randers type. This gives a complete classification of left invariant $(α,β)$-metrics on compact semi-simple Lie groups which are restrictively Clifford-Wolf homogeneous.

math.DG↗

arXiv2013

Clifford-Wolf homogeneous Finsler metrics on spheres

An isometry of a Finsler space is called Clifford-Wolf translation (CW-translation) if it moves all points the same distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous (CW-homogeneous) if for any $x, y\in M$ there is a CW-translation $σ$ such that $σ(x)=y$. We prove that if $F$ is a homogeneous Finsler metric on the sphere $S^n$ such that $(S^n, F)$ is CW-homogeneous, then $F$ must be a Randers metric. This gives a complete classification of CW-homogeneous Finsler metrics on spheres.

math.DG↗

arXiv2013

Killing frames and S-curvature of homogeneous Finsler spaces

In 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.

math.DG↗

arXiv2013

Clifford-Wolf translations of Finsler spaces of negative flag curvature

This paper has been withdrawn by the author due to a crucial sign error in equation 1. An isometry $ρ$ of a connected Finsler space $(M, F)$ is called bounded if the function $d(x, ρ(x))$ is bounded on $M$. It is called a Clifford-Wolf translation if the function $d(x, ρ(x))$ is constant on $M$. In this paper, we prove that on a complete connected simply connected Finsler space of non-positive flag curvature, an isometry is bounded if and only if it is a Clifford-Wolf translation. As an application, we prove that a homogeneous Finsler space of negative flag curvature admits a transitive solvable Lie group of isometries.

math.DG↗

arXiv2012

Ricci-quadratic homogeneous Randers spaces

A Finsler space is called Ricci-quadratic if its Ricci curvature $Ric(x,y)$ is quadratic in $y$. It is called a Berwald space if its Chern connection defines a linear connection directly on the underlying manifold $M$. In this article, we prove that a homogeneous Randers space is Ricci-quadratic if and only if it is of Berwald type.

math.DG↗

arXiv2012

Locally symmetric homogeneous Finsler spaces

Let $(M,F)$ be a connected Finsler space and $d$ the distance function of $(M,F)$. A Clifford translation is an isometry $ρ$ of $(M,F)$ of constant displacement, in other words such that $d(x,ρ(x))$ is a constant function on $M$. In this paper we consider a connected simply connected symmetric Finsler space and a discrete subgroup $Γ$ of the full group of isometries. We prove that the quotient manifold $(M, F)/Γ$ is a homogeneous Finsler space if and only if $Γ$ consists of Clifford translations of $(M,F)$. In the process of the proof of the main theorem, we classify all the Clifford translations of symmetric Finsler spaces.

math.DG↗

arXiv2012

Clifford-Wolf homogeneous Randers spaces

A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the same distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two points $x_1, x_2\in M$ there is a Clifford-Wolf translation $ρ$ such that $ρ(x_1)=x_2$. In this paper, we give a complete classification of connected simply connected Clifford-Wolf homogeneous Randers spaces.

math.DG↗

arXiv2012

Clifford-Wolf translations of Homogeneous Randers spheres

In this paper, we study Clifford-Wolf translations of homogeneous Randers metrics on spheres. It turns out that we can present a complete description of all the Clifford-Wolf translations of all the homogeneous Randers metrics on spheres. The most important point of this paper is that a new phenomena surfaces. Namely, we find that there are some CW-homogeneous Randers spaces which are essentially not symmetric. This is a great difference compared to Riemannian geometry, where any CW-homogeneous Riemannian manifold must be locally symmetric.

math.DG↗

arXiv2012

Clifford-Wolf translations of left invariant Randers metrics on compact Lie groups

A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the sam distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two point $x_1, x_2\in M$ there is a Clifford-Wolf translation $ρ$ such that $ρ(x_1)=x_2$. In this paper, we study Clifford-Wolf translations of left invariant Randers metrics on compact Lie groups. The mian result is that a left invariant Randers metric on a connected compact simple Lie group is Clifford-Wolf homogeneous if and only if the indicatrix of the metric is a round sphere with respect to a bi-invariant Riemannian metric. This presents a large number of examples of non-reversible Finsler metrics which are Clifford-Wolf homogeneous.

math.DG↗

arXiv2012

Clifford-Wolf Translations of Finsler spaces

In this paper, we study Clifford-Wolf translations of Finsler spaces. We first give a characterization of Clifford-Wolf translations of Finsler spaces in terms of Killing vector fields. In particular, we show that there is a natural correspondence between Clifford-Wolf translations and the Killing vector fields of constant length. In the special case of homogeneous Randers spaces, we give some explicit sufficient and necessary conditions for an isometry to be a Clifford-Wolf translation. Finally, we construct some explicit examples to explain some of the results of this paper.

math.DG↗