Related papers: Integral affine 3-manifolds
We consider two classes of smooth maps M^n\to C ^N. Definition. A map f:M^n\to C^N is called an independent map if df_1(p)\wedge...\wedge df_N (p)\neq 0. We are interested in the optimal value of N for all manifolds of dimension n for…
In this paper, we study affine transformations on tori, nilmanifolds and compact abelian groups. For these systems, we show that an equivalent condition for zero entropy is the orbit closure of each point has a nice structure. To be…
We introduce an (equi-)affine invariant diffusion geometry by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to construct an invariant…
The fundamental groups of most (conjecturally, all) closed 3-manifolds with uniform geometries have finite complete rewriting systems. The fundamental groups of a large class of amalgams of circle bundles also have finite complete rewriting…
Given an action of an algebraic torus on a normal affine variety, we describe all open subsets admitting a complete orbit space.
F-manifolds are complex manifolds with a multiplication with unit on the holomorphic tangent bundle with a certain integrability condition. Here the local classification of 3-dimensional F-manifolds with or without Euler fields is pursued.
In this paper we characterize the group of affine transformations of a flat affine simply connected manifold whose developing map is a diffeomorphism. This is proved by making use of some simple facts about homeomorphisms of $\mathbb{R}^n$…
We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry,…
We define an invariant, which we call surface-complexity, of compact 3-manifolds by means of Dehn surfaces. The surface-complexity is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting…
Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to…
In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer,…
We consider closed manifolds that possess a so called rank one ray structure. That is a (flat) affine structure such that the linear part is given by the products of a diagonal transformation and a commuting rotation. We show that closed…
It is shown that in the case of multicomponent integrable systems connected with algebras $A_n$, the discrete transformation $T$ possesses the fine structure and can be represented in the form $T=\prod T_i^{l_i}$, where $T_i$ are n…
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…
A module is called automorphism-invariant if it is invariant under any automorphism of its injective envelope. In this survey article we present the current state of art dealing with such class of modules.
To every real analytic Riemannian manifold M there is associated a complex structure on a neighborhood of the zero section in the real tangent bundle of M. This structure can be uniquely specified in several ways, and is referred to as a…
A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. An algebra is said to be affine complete if every congruence preserving function is a polynomial…
We derive three-dimensional integrable mappings which have two invariants.
We complete the classification of rank two affine manifolds in the moduli space of translation surfaces in genus three. Combined with a recent result of Mirzakhani and Wright, this completes the classification of higher rank affine…
Let $M$ be a compact smooth manifold with corners and $N$ be a finite dimensional smooth manifold without boundary which admits local addition. We define a smooth manifold structure to general sets of continuous mapings $\mathcal{F}(M,N)$…