齐性空间上的循环Finsler度量

Cyclic homogeneous Riemannian manifolds

In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped with a homogeneous spin structure can be viewed as the simplest manifolds after Riemannian symmetric spin spaces. In this paper, we give some characterizations and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds and we obtain the classification of simply-connected cyclic homogeneous Riemannian manifolds of dimension less than or equal to four. We also present a wide list of examples of non-compact irreducible Riemannian $3$-symmetric spaces admitting cyclic metrics and give the expression of…

Cyclic Lorentzian Lie Groups

We consider Lie groups equipped with a left-invariant cyclic Lorentzian metric. As in the Riemannian case, in terms of homogeneous structures, such metrics can be considered as different as possible from bi-invariant metrics. We show that several results concerning cyclic Riemannian metrics do not extend to their Lorentzian analogues, and obtain a full classification of three- and four-dimensional cyclic Lorentzian metrics.

Cyclic metric Lie groups

Cyclic metric Lie groups are Lie groups equipped with a left-invariant metric which is in some way far from being biinvariant, in a sense made explicit in terms of Tricerri and Vanhecke's homogeneous structures. The semisimple and solvable cases are studied. We extend to the general case, Kowalski-Tricerri's and Bieszk's classifications of connected and simply-connected unimodular cyclic metric Lie groups for dimensions less than or equal to five.

具有循环李括号的紧致齐性空间上的均匀加倍性

Uniform doubling on compact homogeneous spaces with cyclic Lie brackets

本文通过循环李括号条件推广Milnor结构，证明了一大类紧致齐性空间上G-不变度量的均匀加倍性，并应用于广义Wallach空间，同时推导了全局Poincaré不等式。

非可解循环度量李代数的结构

Structure of non-solvable cyclic metric Lie algebras

本文系统研究非可解循环度量李代数的结构，引入循环四元组概念分析半单与根式分量的相互作用，并利用双重扩张方法对复数和实数域上既非半单又非可解的非退化情形给出了完整刻画。

Symmetries of Eschenburg spaces and the Chern problem

The known manifolds of positive sectional curvature are either homogeneous spaces or biquotients, i.e. quotients of a compact Lie group by a group acting on the left and right simultaneously. The full isometry group of the homogeneous metrics of positive curvature were determined by K.Shankar. Here we determine the isometry group of some of the biquotients due to Eschenburg and Bazaikin. As an application we obtain, as in the homogeneous case, more counterexamples to the Chern conjecture, which states that an abelian subgroup of the fundamental group of a positively curved manifold is cyclic.