arXiv research

Almost Zoll affine surface found on cylinder.

We study affine maps between affine manifolds. Even when the fibers are compact and diffeomorphic, two of them can inherit different affine structures from the source space. This leads to a fixed linear holonomy deformation theory of the affine structure of an affine manifold. We found various conditions which make the…

In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-var…

In this paper, we show that a compact affine manifold endowed with an Affine Anosov transformation is finitely covered by a complete affine nilmanifold.

An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with holonomy consisting of affine transformati…

An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting o…

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for $L_φ$ affine surface areas are established.

Study on completeness in affine and statistical geometry.

This thesis is devoted to the study of affine processes and their applications in financial mathematics. In the first part we consider the theory of time-inhomogeneous affine processes on general state spaces. We present a concise setup for time-inhomogeneous Markov processes. For stochastically continuous affine proce…

In this paper, we consider affine self-similar solutions for the affine curve shortening flow in the Euclidean plane. We obtain the equations of all affine self-similar solutions up to affine transformations and solve the equations or give descriptions of the solutions for the degenerate case. Some new special solution…

A (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space $\mathbb{R}^3$ with transition maps in the affine transformation group $\mathrm{Aff}(\mathbb{R}^3)$. We will show that a connected closed affine $3$-manifold is either an affine Hopf $3$-manifold or decomposes canonically to conca…

We study affine Jacobi structures on an affine bundle $π:A\to M$, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on $A$ and Lie algebroid structures on the vector bundle $A^+=\bigcup_{p\in M}Aff(A_p,\R)$ of affine functionals. Som…

Paper shows affine constraint is unnecessary for high-dimensional data.

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygo…

An affine Cartan calculus is developed. The concepts of special affine bundles and special affine duality are introduced. The canonical isomorphisms, fundamental for Lagrangian and Hamiltonian formulations of the dynamics in the affine setting are proved.

Calculates affine transformations for specific homogeneous spaces.

In this paper, we introduce a new structure, namely, affine Szabó connection. We prove that, on $2$-dimensional affine manifolds, the affine Szabó structure is equivalent to one of the cyclic parallelism of the Ricci tensor. A characterization for locally homogeneous affine Szabó surface is obtained. Examples of two- a…

In this paper, we introduce a new parameter, the affine twist parameter for the affine deformation of a sphere with holes. We show that the affine deformation space can be parametrized by Margulis invariants and affine twist parameters. The affine twist parameter is canonically regarded as a correspondence to the Fench…

The paper studies a new class of affine maximal surfaces with singularities.

Geometric models for algebraic suspensions using affine deformation spaces.

We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a …

Overview of affine surface area and its history.

Paper proves certain closed affine manifolds without invariant lines don't exist.

Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this te…

A branched affine structure on a compact topological surface with marked points is a complex affine structure outside the marked points. We give a proof of an unpublished foundational theorem of Veech, stating that any branched affine surface can be decomposed into affine triangles and some annulus-shaped cylinders. Th…

New complete hypersurfaces found in affine geometry.

New methods classify convex lattice polygons for affine dimers.

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 L…

Yau's Affine Normal Descent optimizes smooth unconstrained problems with geometrically adapted directions.

An (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space ${\mathbf R}^3$ with transition maps in the affine transformation group $Aff({\mathbf R}^3)$. Equivalently an affine $3$-manifold is a $3$-manifold with a flat torsion-free affine connection. We show that a closed affine $3$-mani…

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 …

In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines …

Affine deformations of convex cones yield special spacetime structures.

The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes…

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. Th…

Survey on affine Anosov representations and their implications.

In this paper we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting times of jumps can be both inaccessible and predictable. To this end we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the co…

Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure …

Affine cactus groups are CAT(0) and hyperbolic.

A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles hav…

New theorem extends Hadamard's to transversely affine geometry.

Partial answer to affineness of entire Grauert tubes, with Stein manifold criterion.

Affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $(p,q)$-tensors are studied. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields has a natural graded strict Lie 2-algebra stru…

This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connecti…

The class of affine LIBOR models is appealing since it satisfies three central requirements of interest rate modeling. It is arbitrage-free, interest rates are nonnegative and caplet and swaption prices can be calculated analytically. In order to guarantee nonnegative interest rates affine LIBOR models are driven by no…

The paper explores fully affine maximal curves and their properties.

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field (sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\widetilde{J}$-ta…

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups ${\rm GA}(2)={\rm GL}(2,{\bf R})\ltimes {\bf R}^2$ and ${\rm GA}(3)={\rm GL}(3,{\bf R})\ltimes {\bf R}^3$, respectively. We define general-affine length parameter and curvatures and show how such …