On projective group properties of the 6d h-space of the type [33]

Projective Hessian and Sasakian manifolds

The Hessian geometry is the real analogue of the Kähler one. Sasakian geometry is an odd-dimensional counterpart of the Kähler geometry. In the paper, we study the connection between projective Hessian and Sasakian manifolds analogous to the one between Hessian and Kähler manifolds. In particular, we construct a Sasakian structure on $TM\times \mathbb{R}$ from a projective Hessian structure on $M$. Especially, we are interested in the case of invariant structure on Lie groups. We define semi-Sasakian Lie groups as a generalization of Sasakian Lie groups. Then we construct a semi-Sasakian struc…

Invariant metrizability and projective metrizability on Lie groups and homogeneous spaces

In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of thes…

五维幂零李群上的仿射与射影向量场

Affine and Projective vector fields on five-dimensional nilpotent Lie groups

本文基于左不变度量分类，对五维单连通黎曼幂零李群上的左不变仿射与射影向量场进行了完整分类，证明了所有射影向量场必然是仿射的。

射影Banach李双代数、射影Yang-Baxter方程与射影Poisson-Lie群

Projective Banach Lie bialgebras, the projective Yang-Baxter equation and projective Poisson-Lie groups

本文引入射影Banach李双代数、射影Yang-Baxter方程及射影Poisson-Lie群，证明拟三角射影r-矩阵产生射影Banach李双代数，射影Poisson-Lie群微分具有该结构，并给出三角射影r-矩阵的等价描述。

Projectively invariant objects and the index of the group of affine transformations in the group of projective transformations

The paper is grown from the lecture course "Metric projective geometry" which I hold at the summer school "Finsler geometry with applications" at Karlovassi, Samos, in 2014, and at the workshop before the 8th seminar on Geometry and Topology of the Iranian Mathematical society at the Amirkabir University of Technology in 2015. The goal of this lecture course was to show how effective projectively invariant objects can be used to solve natural and named problems in differential geometry, and this paper also does it: I give easy new proofs to many known statements, and also prove the following n…

Virtually abelian Kähler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral symplectic representations.

On discrete projective transformation groups of Riemannian manifolds

We prove rigidity facts for groups acting on pseudo-Riemannian manifolds by preserving unparameterized geodesics.

Low dimensional Lie groups admitting left invariant flat projective or affine structures

We prove that any real Lie group of dimension \leq 5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension \leq 5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.

A flat torus theorem for convex co-compact actions of projective linear groups

In this paper, we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well known flat torus theorem for ${\rm CAT}(0)$ spaces.

Characterizing the unit ball by its projective automorphism group

In this paper we study the projective automorphism group of domains in real, complex, and quaternionic projective space and present two new characterizations of the unit ball in terms of the size of the automorphism group and the regularity of the boundary.

Projective unitary representations of smooth Deligne cohomology groups

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations is finite, and is determined by the cohomology of the manifold.

Projective superflows. II. $O(3)$ and the icosahedral group

Let $X\in\mathbb{R}^{n}$. For $ϕ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $ϕ^{t}=t^{-1}ϕ(Xt)$. A projective flow is a solution to the projective translation equation $ϕ^{t+s}=ϕ^{t}\circϕ^{s}$, $t,s\in\mathbb{R}$. The projective superflow is a projective flow with a rational vector field which, among projective flows with a given symmetry, is in a sense unique and optimal. In this second part we classify $3$-dimensional real superflows. Apart from the superflow $ϕ_{\hat{\mathbb{T}}}$ (with a group of symmetries being all symmetries of a tetrahedron) and the superflow $ϕ…

On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

We prove that for any affine variety S defined over Q there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety Hom(G, PO(3))//PO(3). The subset U contains all real points of S . As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties.

在凸实射影几何中相对超几何群的结构

The structure of relatively hyperbolic groups in convex real projective geometry

本文研究了在凸实射影几何中相对超几何群的结构，通过凸紧致作用和不变凸子集的特征化，建立了与CAT(0)空间结果的类比，并给出了边界之间的等距同胚。

The mapping class group of manifolds which are like projective planes

In this paper we compute the mapping class group of simply-connected closed smooth manifolds $M$ with integral homology $H_{*}(M) \cong \mathbb Z \oplus \mathbb Z \oplus \mathbb Z$ provided that $\dim M \ne 4$.