arXiv research

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…

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…

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 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…

This paper describes integral affine structures on compact 3-manifolds.

Characterizes affine vector fields on Finsler manifolds with rigidity results.

Characterizes flat affine connections on manifolds.

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

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

Completed classification of rank 2 affine manifolds in genus 3.

The paper characterizes flat affine connections on manifolds and Lie groups.

Chern's conjecture on affine manifolds is proven to be true.

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…

We solve a linear equation on affine manifolds, finding finite-dimensional solutions.

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…

Affine 3-manifolds with centralizing holonomy are complete.

In this paper, we consider the cyclic parallel Ricci tensor condition, which is a necessary condition for an affine manifold to be Szabó. We show that, in dimension $3$, there are affine manifolds which satisfy the cyclic parallel Ricci tensor but are not Szabó. Conversely, it is known that in dimension $2$, the cyclic…

Study geodesic and affine Killing completeness in homogeneous affine surfaces.

Study on completeness in affine and statistical geometry.

A Sim(n-1,1) affine manifold is an affine manifold whose linear holonomy is contained in the similarity lorentzian group but not in the lorentzian group. The class of similarity lorentzian affine manifolds is a small part in the nice class of conformally lorentzian flat manifolds. In this paper we show that a compact S…

The study examines closed manifolds with ray nil-affine structures and their completeness.

Study shows complete affine manifolds have zero simplicial volume.

The paper explores properties of affine Szabó manifolds and their metrics.

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

The paper constructs new structures for manifolds using connections and combinations.

Affine manifolds linked to integrable equations and geometric structures.

We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections.

New manifold structures on Weyl group orbit spaces proven.

Equi-affine curvature of curves in 2-manifolds linked to Frenet curvature.

The connection between Jacobi fields and odular structures of affine manifold is established. It is shown that the Jacobi fields generate the natural geoodular structure of affinely connected manifolds.

The paper explores equi-affine curvatures in pseudo-Riemannian manifolds.

Study of superintegrable systems linked to affine hypersurfaces.

Geodesic connectedness proved for statistical manifolds with divisible cubic forms.

Paper uses flow to prove theorem on Higgs bundles.

The paper constructs submanifolds with corners in Delzant polytopes from affine subspaces.

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 i…

The paper studies affine manifolds with linear foliations and their topological properties.

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold $M$ with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to $\bR^3$. Hence, $M$ is irreducib…

In this paper we prove that for a complete, connected and oriented Käler affine manifold $(M,G)$ of dimension $n,$ if it is Kähler affine Ricci flat or the K$\ddot{a}$hler affine scalar curvature $S\equiv0,$ ($n\leq 5$), then the universal covering manifold $\widetilde{M}$ of $M$ is isometric to the Euclidean n-space $…

The paper classifies compact affine quaternionic curves and surfaces.

This is a review with examples concerning the concepts of affine (in particular, constant and linear) vector fields and fundamental vector fields on a manifold. The affine, linear and constant vector fields on a manifold are shown to be in a bijective correspondence with the fundamental vector fields on it of respectiv…

New conditions ensure geodesic connectedness of affine manifolds.

We introduce the notion of affine Legendrian submanifolds in Sasakian manifolds and define a canonical volume called the $φ$-volume as odd dimensional analogues of affine Lagrangian (totally real or purely real) geometry. Then we derive the second variation formula of the $φ$-volume to obtain the stability result in so…

The study classifies product affine hyperspheres in R^n+1.

Constructs generalized Frobenius manifolds for specific Weyl groups.

Study of manifolds with flat connections and diagonal metrics leading to vanishing Euler characteristic.

Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair (E, φ), consisting of a flat vector bundle E over M and a flat nonzero section φ of E, admits a solution to the vortex equation if and only if it is polystable. To prove this, we adapt the dimensional redu…

Study of affine and projective structures on foliated complex manifolds.