Calculates affine transformations for specific homogeneous spaces.
problem Computing groups of affine transformations on homogeneous spaces.
method Analyzes conditions for affine connections and uses them to establish group isomorphisms.
result Groups of affine transformations are locally isomorphic under specified conditions.
Corrected proof of geodesics in homogeneous Finsler spaces using affine method.
problem Proving the existence of homogeneous geodesics in homogeneous Finsler spaces.
method Used affine method to correct the proof of the algebraic method.
result Correctly proved that any homogeneous Finsler space admits a homogeneous geodesic through any point.
A moduli space is constructed for affine homogeneous spaces using their local geometry.
problem Describing and classifying affine homogeneous spaces locally and globally.
method Local description via curvature, torsion, and connection; algebraic variety construction; Spencer cohomology for infinitesimal deformations.
result A moduli space M(glV) is constructed for affine homogeneous spaces of dimension dimV. Study geodesic and affine Killing completeness in homogeneous affine surfaces.
problem Geodesic and affine Killing completeness in homogeneous affine surfaces.
method Examined using the solution space of the quasi-Einstein equation.
result Characterized geodesic and affine Killing completeness in homogeneous affine surfaces.
Study solutions to affine quasi-Einstein equation on homogeneous surfaces.
problem Understanding solutions to affine quasi-Einstein equation on homogeneous surfaces.
method Using the dimension of affine Killing vector fields as a framework, we provide explicit descriptions of solution spaces.
result Very explicit descriptions of solution spaces for gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein manifolds.
Study the topology of spaces of locally homogeneous affine surfaces.
problem Classify and analyze the topology of spaces of locally homogeneous affine manifolds.
method Examine spaces of locally homogeneous affine manifolds arising from Opozda's classification, determine their topology based on the Ricci tensor and group type.
result Determine the topology of spaces of Type A and B models, relating it to the Ricci tensor and group type.
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.
problem Understanding parallel transport in reductive homogeneous spaces with torsion.
method Generalizing previous results on affine symmetric spaces, proving compactness of shape operators, and proposing definitions for regularized mean curvatures.
result Each fiber of the parallel transport map over a reductive homogeneous space is minimal in both senses.
Researchers discover all affinely homogeneous models for surfaces in 4D space.
problem Identifying all affinely homogeneous models for surfaces in 4D space.
method Improved power series method of equivalence, capturing invariants at the origin, creating branches, and infinitesimalizing calculations.
result Find several inequivalent terminal branches yielding each to some nonempty moduli space of homogeneous models.
Classifies tube domains with specific properties in complex spaces.
problem Classifying tube domains with unique geometric properties.
method Analyzes tube domains in complex spaces with homogeneous boundaries and specific Levi forms.
result Classifies tube domains with large automorphism groups in arbitrary dimensions.
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.
problem Characterize moduli spaces of homogeneous affine connections on surfaces.
method Analyze Type A and B surfaces, compute invariants, and examine moduli space structures.
result Moduli spaces for Type B surfaces are simply connected 4-manifolds with Betti number 1.
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.
problem Characterizing invariant affine connections on 3-Sasakian homogeneous manifolds.
method Explicit construction and analysis of all 3-Sasakian homogeneous manifolds.
result Unique 3-Sasakian homogeneous manifolds admit nontrivial Einstein connections with skew-torsion. Geodesically equivalent metrics on homogeneous spaces are shown to be affinely equivalent.
problem Characterizing geodesically equivalent metrics on homogeneous spaces.
method Analyzing left G-invariant metrics on G/H and using algorithms to find geodesically equivalent metrics. result Existence of non-proportional geodesically equivalent metrics implies non-full holonomy algebra.
Study on moduli spaces of Type~B surfaces with torsion.
problem Examining moduli spaces of locally homogeneous surfaces with torsion.
method Analyzing symmetric Ricci tensor and affine Killing vector fields.
result Determined the space of affine Killing vector fields.
Study classifies special Hessian rank 2 hypersurfaces in 4D space.
problem Classifying hypersurfaces with constant Hessian rank 2.
method Power series method of equivalence, Lie's classification spirit.
result 34 inequivalent terminal branches, each with a nonempty moduli space.
The paper finds a correspondence between invariant PDEs and hypersurfaces.
problem Constructing invariant PDEs on projective and affine spaces.
method Applying a general method to specific cases of projective and affine spaces.
result Projectively or affinely invariant PDEs correspond to CO(d, n-d) invariant 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…
Classifies homogeneous affine surfaces and classifies gradient Ricci solitons.
problem Classifying homogeneous affine surfaces and their properties.
method Examined Lie algebra of affine Killing vector fields and classified gradient Ricci solitons.
result Complete classification of homogeneous affine gradient Ricci solitons.
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.
problem Classifying locally homogeneous affine surfaces with torsion.
method Direct analysis of affine Killing equations.
result New classification of locally homogeneous affine surfaces with torsion.
Invariant connections on odd spheres described.
problem Characterizing invariant affine connections on odd-dimensional spheres.
method Using homogeneous spaces and invariant tensors, the paper describes all connections on spheres.
result Unique odd-dimensional spheres S3 and S7 admit nontrivial invariant connections. 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.
problem Effects of torsion on affine Killing vectors on homogeneous surfaces.
method Complete description of Lie algebras of affine Killing vector fields on homogeneous surfaces.
result Complete description of Lie algebras of affine Killing vector fields on homogeneous surfaces.
New families of complete hyperbolic affine spheres are constructed from Hildebrand's semi-homogeneous cones.
problem Constructing families of complete hyperbolic affine spheres from semi-homogeneous cones.
method Computed isothermal parametrizations, affine metrics, and cubic forms for Hildebrand's examples; constructed associated families using Weierstrass functions.
result Generic members of the constructed affine spheres are given by Weierstrass functions.
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.
problem Characterize homogeneous affine surfaces with Hessian rank 2.
method Investigate algebra of differential invariants under affine transformation group.
result Organize homogeneous models into inequivalent branches.
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. …
Study of time-inhomogeneous affine processes in finance.
problem Understanding and modeling financial processes with time-varying parameters.
method Developed a theory for time-inhomogeneous affine processes and applied it to financial market models.
result Affine processes can be modified to include real-valued processes, improving model flexibility.
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 C1 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.
problem Defining and characterizing affine Szabó connections on manifolds.
method Introduced affine Szabó connections and proved their equivalence to cyclic parallelism on 2D affine manifolds.
result Characterization of locally homogeneous affine Szabó surfaces.
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.
problem Solving a linear equation on affine manifolds.
method Proving the space of solutions is finite-dimensional and characterizing the maximal dimension.
result The maximal dimension of solutions is achieved only on strongly projectively flat manifolds.
Study of symmetric affine surfaces with non-vanishing torsion.
problem Characterizing geometries of symmetric affine surfaces with torsion.
method Complete classification of local geometries assuming parallel torsion and local homogeneity for non-parallel torsion.
result Classification of local geometries for symmetric affine surfaces with torsion, generalizing previous results.
The paper classifies invariant connections and Einstein structures on isotropy irreducible spaces.
problem Classifying invariant connections and Einstein structures on isotropy irreducible spaces.
method Systematic study and classification of invariant affine or metric connections on naturally reductive spaces.
result Classification of invariant metric connections with skew-torsion and abla-Einstein structures. 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.
problem Understanding invariant covariant derivatives on homogeneous spaces.
method Expressing covariant derivatives in terms of horizontally lifted vector fields and bilinear maps.
result Existence and characterization of invariant covariant derivatives.
The paper classifies and determines properties of specific geometric structures.
problem Classifying and understanding affine reductive homogeneous manifolds.
method Analyzing Lie groups and their homogeneous spaces, using sharply transitive sections.
result Classification of left A-loops and determination of their properties.
We study affine Jacobi structures on an affine bundle π:A→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+=⋃p∈MAff(Ap,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 ∇ be …
No non-product Hessian rank 1 affine homogeneous hypersurfaces exist in dimensions 5 and above.
problem Identifying non-product Hessian rank 1 affine homogeneous hypersurfaces in higher dimensions.
method Developed a normal form for hypersurfaces under the affine group, up to order ≤ n+5, in any dimension n ≥ 2.
result Non-existence of non-product Hessian rank 1 affine homogeneous hypersurfaces 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.
problem Characterizing holomorphic structures on Oeljeklaus-Toma manifolds.
method Proving local homogeneity for various holomorphic geometric structures.
result Holomorphic geometric structures on Oeljeklaus-Toma manifolds are locally homogeneous.
New types of Einstein manifolds discovered from homogeneous surfaces.
problem Understanding quasi-Einstein equations on homogeneous surfaces.
method Modified Riemannian extension and warped product construction.
result Discovery of new Einstein manifolds.