arXiv research

Characterizes flat affine symplectic Lie groups and their properties.

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…

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…

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

The paper explores flat affine and symplectic structures on Lie groups.

Characterizes flat affine connections on manifolds.

A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.

Notes on flat pseudo-Riemannian manifolds, focusing on their characterization and properties.

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.

Study classifies half conformally flat GQE manifolds of signature (2,2).

The aim of this paper is to give a local description of affine surfaces, whose induced Blaschke structure is projectively flat. We show that such affine surfaces with constant Gauss affine curvature and indefinite induced Blaschke metric are described by soliton equations.

Calculates affine transformations for specific homogeneous spaces.

Paper presents a new flat triangular form for systems.

Study examines geodesic completeness of affine surfaces using quasi-Einstein equations.

Study shows dynamics of rank 1 orbifolds in flat surfaces.

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…

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…

Constructs Bach flat manifolds using modified Riemannian extension.

The results of the paper concern the topological structure of complete riemannian manifolds with cyclic holonomy groups and low-dimensional orientable complete flat manifolds. We also discuss related results such as the affine classification of orientable complete flat 4-manifolds, an algebraic criterion of an affine e…

Paper presents a new triangular form for flat systems.

Kunneth formula derived for flat affine manifolds and applied to Hessian metrics.

Paper uses flow to prove theorem on Higgs bundles.

We investigate the geometric properties of hyperbolic affine flat, affine minimal surfaces in the equiaffine space $\mathbb{A}^3$. We use Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of such surfaces and…

Study of convex hypersurfaces with specific curvature properties.

We prove Chern conjecture, which states that the Euler characteristic vanishes for closed flat affine manifolds. Our key innovation is a deformation argument for the Euler form.

The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this articl…

Flat affine subvarieties found in OT-manifolds.

New complex manifolds found with flat structure.

The paper classifies fibrations of 3-dimensional flat orbifolds.

Kahler toric manifolds linked to dually flat spaces via affine isometry.

The paper studies the affine focal set of submanifolds in hypersurfaces, describing conditions for regularity and singularities.

Study the geometry of a Lie group using Hessian and flat affine structures.

This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.

Study parallel tensors on affine surfaces to characterize Ricci recurrence.

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.

New proof shows affine manifolds with parallel volume are Riemannian-flat.

New polynomials show all knots of a certain type have simple structure.

On an affine flat manifold with coordinates x^j and convex local potential function f, we call the affine Kahler metric f_{ij} dx^i dx^j semi-flat Calabi-Yau if it satisfies det f_{ij} = 1. Recently Gross-Wilson have constructed many such metrics on S^2 minus 24 singularities, as degenerate limits of Calabi-Yau metrics…

Let (E, \varphi) be a flat Higgs bundle on a compact special affine manifold M equipped with an affine Gauduchon metric. We prove that (E, \varphi) is polystable if and only if it admits an affine Yang-Mills-Higgs metric.

The paper computes KV cochain differentials and their geometric implications.

We show that the total space of any affine $\mathbb{C}$-bundle over $\mathbb{CP}^1$ with negative degree admits an ALE scalar-flat Kähler metric. Here the degree of an affine bundle means the negative of the self-intersection number of the section at infinity in a natural compactification of the bundle, and so for line…

Study jets of flat partial connections in foliations.

We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence o…

We study the real Monge-Ampère equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holomorphic…

Affirmative answer to flat holomorphic Cartan geometries on complex tori.

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat princip…

The paper explores conditions for statistical manifolds to be immersed and dual to each other.

Analyzes limits of flat metrics on complex curves.