Affine Anosov transformation of affine manifolds

Quasi-Fuchsian co-Minkowski manifolds

This survey is an introduction to the geometry of co-Minkowksi space, the space of unoriented spacelike hyperplanes of the Minkowski space. Affine deformations of cocompact lattices of hyperbolic isometries act on it, in a way similar to the way that quasi-Fuchsian groups act on hyperbolic space. In particular, there is a convex core. There is also a unique "mean" hypersurface, i.e. with traceless second fundamental form. The mean distance between the mean hypersurface and the lower boundary of the convex core endows the space of affine deformations of a given lattice with an asymmetric norm. …

Classification of cubics up to affine transformations

Classification of cubics (that is, third order planar curves in the $R^2$ up to certain transformations is interested since Newton, and treated by several authors. We classify cubics up to affine transformations, in seven class, and give a complete set of representatives of the these classes. This result is complete and briefer than the similar results.

Flat Affine Manifolds And Their Transformations

We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine $n$-dimensional manifold, acts on $\mathbb{R}^n$ leaving an open orbit when its dimension is greater than $n$. Moreover, when the dimension of the group of aff…

正表示的真仿射形变

Proper affine deformations of positive representations

本文为自由群到特殊正交群的正 Anosov 表示构造上循环，诱导真仿射作用，并建立基本域，证明商流形同胚于柄体。

Affine transformations and parallel lightlike vector fields on compact Lorentzian $3$-manifolds

We describe the compact Lorentzian $3$-manifolds admitting a parallel lightlike vector field. The classification of compact Lorentzian $3$-manifolds admitting non-isometric affine diffeomorphisms follows, together with the complete description of these morphisms. Such a Lorentzian manifold is in some sense an equivariant deformation of a flat one.

Flat Affine and Symplectic Geometries on Lie Groups

In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of Lie groups between the group of affine transformations of a flat affine and simply connected manifold and the classical group of affine transformations of $\mathbb{R}^n$. In the last section, for each flat left invariant affine symplectic connection on the group of affine transfor…

Flat Affine or Projective Geometries on Lie Groups

This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. These groups could play an essential role in the study of homogeneous spaces $M=G/H$ admitting flat affine or flat projective structures invariant under the natural action of $G$ on $M$. A. Medina asked several years ago if the group of affine transformations of a flat affine Lie g…

Affine Structures on Jet and Weil Bundles

Weil algebra morphism induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cases, this structure of affine bundle is passed down to Jet spaces. We give a characterization of this fact in algebraic terms. This algebraic condition also determines an affine structure between the groups of automorphisms of related Weil algebras

具有离散迷向群的齐性空间的仿射变换群

The group of affine transformations of homogeneous spaces with discrete isotropy

针对离散迷向群情形，本文给出计算齐性 $G$-空间仿射变换群的方法，并证明相关仿射变换群局部同构，还应用于平坦仿射曲面与三维环面。

Affine vector fields on Finsle manifolds

We give characterizations of affine transformations and affine vector fields in terms of the spray. By utilizing the Jacobi type equation that characterizes affine vector fields, we prove some rigidity theorems of affine vector fields on compact or forward complete non-compact Finsler manifolds with non-positive total Ricci curvature.

Linear Transformations on Affine-Connections

We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We show that the initial functional is invariant under the aforementioned group of transformations iff its $Γ$-variation produces tensor of a given symmetry. Conversely if the tensor produced by the $Γ$-variation of the functional respects a certain symmetry then the functional is …

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…

Geometric Properties of Conformal Transformations on $\mathbb{R}^{p,q}$

We show that conformal transformations on the generalized Minkowski space $\mathbb{R}^{p,q}$ map hyperboloids and affine hyperplanes into hyperboloids and affine hyperplanes. We also show that this action on hyperboloids and affine hyperplanes is transitive when $p$ or $q$ is $0$, and that this action has exactly three orbits if $p, q \ne 0$. Then we extend these results to hyperboloids and affine planes of arbitrary dimension. These properties generalize the well-known properties of Möbius (or fractional linear) transformations on the complex plane $\mathbb{C}$.

Riemannian Geometry Approach for Minimizing Distortion and its Applications

Given an affine transformation $T$, we define its Fisher distortion $Dist_F(T)$. We show that the Fisher distortion has Riemannian metric structure and provide an algorithm for finding mean distorting transformation -- namely -- for a given set $\{T_{i}\}_{i=1}^N$ of affine transformations, find an affine transformation $T$ that minimize the overall distortion $\sum_{i=1}^NDist_F^{2}(T^{-1}T_{i}).$ The mean distorting transformation can be useful in some fields -- in particular, we apply it for rendering affine panoramas.

On the topology of compact affine manifolds

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold admits a flat, symmetric and complete connection. If the completeness assumption is dropped, the manifold is not necessarily obtained as the quotient of the Euclidean space through a properly discontinuous group of affine transformations. In fact the universal cover may no longer be the Euclidean space. The main result of this paper states that all compact aff…

仿射变换离散群的真作用

Proper actions of discrete groups of affine transformations

本文围绕Milnor1977年提出的问题，介绍Margulis关于自由群三维真仿射作用存在性的突破，梳理相关历史背景、领域最新进展及该发现启发的未来研究方向。