arXiv research

Calculates affine transformations for specific homogeneous spaces.

Corrected proof of geodesics in homogeneous Finsler spaces using affine method.

A moduli space is constructed for affine homogeneous spaces using their local geometry.

Study geodesic and affine Killing completeness in homogeneous affine surfaces.

Study solutions to affine quasi-Einstein equation on homogeneous surfaces.

Study the topology of spaces of locally homogeneous affine surfaces.

We classify the non-degenerate homogeneous hypersurfaces in real and complex affine four-space whose symmetry group is at least four-dimensional.

Parallel transport map over reductive spaces is an affine submersion.

Researchers discover all affinely homogeneous models for surfaces in 4D space.

Classifies tube domains with specific properties in complex spaces.

A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous sp…

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear …

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members…

Study moduli spaces of non-flat connections on surfaces.

The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the function. In particular, the nonzero level sets of a homogeneous polynomial are proper…

The study describes invariant affine connections on 3-Sasakian homogeneous manifolds.

Geodesically equivalent metrics on homogeneous spaces are shown to be affinely equivalent.

Study on moduli spaces of Type~$\mathcal{B}$ surfaces with torsion.

Study classifies special Hessian rank 2 hypersurfaces in 4D space.

The paper finds a correspondence between invariant PDEs and hypersurfaces.

Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when…

The purpose of these survey notes is to give a presentation of a classical theorem of Nomizu that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras.

New proof for homogeneous affine surfaces with torsion.

For k at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on that of an indecomposible symmetric space. All the local scalar Weyl curvature i…

Study torsion's impact on 2D affine Killing vectors on homogeneous surfaces.

We provide a new proof for regularity of affine processes on general state spaces by methods from the theory of Markovian semimartingales. On the way to this result we also show that the definition of an affine process, namely as stochastically continuous time-homogeneous Markov process with exponential affine Fourier-…

Study on affine surfaces with specific algebraic properties.

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 previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. …

In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least $C^1$ boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional par…

New affine Szabó connections defined on smooth manifolds.

We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.

We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an alternate proof of the main result of [13], showing that the characterization of…

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

The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of t…

A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann-Cartan space is introduced in a natura…

Study of symmetric affine surfaces with non-vanishing torsion.

The paper classifies invariant connections and Einstein structures on isotropy irreducible spaces.

We give an algebraic characterization of the possible characteristic tensors of an infinitesimally homogeneous affine manifold with G-structure.

Invariant covariant derivatives on homogeneous spaces are characterized.

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…

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let $\nabla$ be …

No non-product Hessian rank 1 affine homogeneous hypersurfaces exist in dimensions 5 and above.

We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies in a local result that cla…

Holomorphic structures on Oeljeklaus-Toma manifolds are shown to be locally homogeneous.

New types of Einstein manifolds discovered from homogeneous surfaces.

We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisation…

Left-symmetric algebras help define affine spheres.