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Shaoqiang Deng

Publications and source records attributed to Shaoqiang Deng.

At least 19 recordsLinked to original sources
arXiv2026

A diameter gap for irreducible compact symmetric spaces of fixed rank

For every positive integer $r$, we prove that positive-diameter isometric quotients of simply connected irreducible compact Riemannian symmetric spaces of rank $r$ have a uniform positive lower bound on their normalized diameter. The bound is independent of dimension and of the acting group, which may be disconnected. The rank-one case is due to Gorodski, Lange, Lytchak, and Mendes \cite{GLLM}. For the higher-rank Grassmannian families, we construct nonconstant invariant functions of bounded trigonometric degree. The real and complex cases use a moment map argument. The quaternionic case uses a quartic rigidity argument and the classification of compact quaternionic symmetric spaces.

math.DG↗
arXiv2025

A counter example for the Homogeneity Conjecture

We construct a counter example to show that the Homogeneity Conjecture, first proposed by J.A. Wolf in 1962, is not true. To be precise, we prove that on the Lie group Sp(2), there exists a left invariant Riemannian metric and a cyclic subgroup Γ of order (2n+1), such that the left translation of each element of Γ on Sp(2) is a Clifford-Wolf translation, but the Riemannian quotient Γ\Sp(2) is not homogeneous.

math.DG↗
arXiv2023

Polyubles, Poisson homogeneous spaces and multi-flag varieties

A polyuble of a Manin triple can be regarded as the ``$n$-th power'' of it, which plays an important rule in the study of Poisson geometry, mathematical physics and Lie theory. In this paper, we first construct an isomorphism between the $mn$-ble and the $n$-ubles of $m$-uble by colored graph and point out it is unique. Then, we construct a class of Poisson homogeneous spaces and obtain a class of Poisson homeomorphisms between them based on the first main result. Last, we apply first two main results to multi-flag varieties as well as multi-double flag varieties and construct a class of global Poisson isomorphisms between them as well as their $T$-leaves.

math-ph↗
arXiv2018

Geodesic orbit Finsler metrics on Euclidean spaces

A Finsler space $(M,F)$ is called a geodesic orbit space if any geodesic of constant speed is the orbit of a one-parameter subgroup of isometries of $(M, F)$. In this paper, we study Finsler metrics on Euclidean spaces which are geodesic orbit metrics. We will show that, in this case $(M, F)$ is a fiber bundle over a symmetric Finsler space $M_1$ of non-compact type such that each fiber $M_2$ is a totally geodesic nilmanifold with a step-size at most 2, and the projection $π:M\rightarrow M_1$ is a Finslerian submersion. Furthermore, when $M_1$ has no Hermitian symmetric factors, the fiber bundle description for $M$ can be strengthened to $M=M_1\times M_2$ as coset spaces, such that each product factor is totally geodesic in $(M,F)$ and is a geodesic orbit Finsler space itself. Finally, we use the techniques in this paper to discuss the interaction between the geodesic orbit spaces and the negative (non-positive) curved conditions, and provide new proofs for some of our previous results.

math.DG↗
arXiv2017

Homogeneous Finsler spaces and the flag-wise positively curved condition

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) Condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank $2$ non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space $S^3\times \mathbb{R}$ which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on $S^2\times S^3$ and $S^6\times S^7$ which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on $S^3\times S^3$, shedding some light on the long standing general Hopf conjecture.

math.DG↗
arXiv2017

Geodesic and curvature of piecewise flat Finsler surfaces

A piecewise flat Finsler metric on a triangulated surface $M$ is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex. Using the edge-crossing equation, we define two classes of piecewise flat Finsler surfaces, namely, Landsberg type and Berwald type. We deduce an explicit condition for a geodesic to be extendable at a vertex, and define the curvature which measures the \textit{amount} of such extensions. The dependence of the curvature on an incoming or outgoing tangent direction corresponds to the feature of flag curvature in Finsler geometry. When the piecewise flat Finsler surface is of Landsberg type, the curvature is only relevant to the vertex, and we prove a combinatoric Gauss-Bonnet formula which generalizes both the Gauss-Bonnet formulas for piecewise flat Riemannian manifolds and for smooth Landsberg surfaces.

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arXiv2017

Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds

It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds is presented. In particular, we show that on the standard homogeneous Einstein manifolds, except for some special cases, there exist plenty of such metrics. A further interesting result of this paper is that on some compact semisimple Lie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics which are not product metrics.

math.DG↗
arXiv2016

Locally 2-fold symmetric manifolds are locally symmetric

A manifold is locally \emph{$k$-fold symmetric}, if for any point and any $k$-dimensional vector subspace tangent to this point there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that $k$-dimensional vector subspace is minus the identity. We show that for $k \ge 2$, Riemannian, pseudoriemannian and Finslerian locally $k$-fold symmetric manifolds are locally symmetric.

math.DG↗
arXiv2016

Rigidity of negatively curved geodesic orbit Finsler spaces

We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.

math.DG↗
arXiv2015

Towards the classification of odd dimensional homogeneous reversible Finsler spaces with positive flag curvature

In this paper, we use the flag curvature formula for homogeneous Finsler spaces in our previous work to classify odd dimensional smooth coset spaces admitting positively curved reversible homogeneous Finsler metrics. We will show that the most features of L. Bérard-Bergery's classification results for odd dimensional positively curved Riemannian homogeneous spaces can be generalized to reversible Finsler spaces.

math.DG↗
arXiv2015

Even dimensional homogeneous Finsler spaces with positive flag curvature

In this paper, we use the technique of Finslerian submersion to deduce a flag curvature formula for homogeneous Finsler spaces. Based on this formula, we give a complete classification of even-dimensional smooth coset spaces $G/H$ admitting $G$-invariant Finsler metrics with positive flag curvature. It turns out that the classification list coincides with that of the even dimensional homogeneous Riemannian manifolds with positive sectional curvature obtained by N.R. Wallach. We also find out all the coset spaces admitting invariant non-Riemannian Finsler metrics with positive flag curvature.

math.DG↗
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.

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