Study first BGG operators on homogeneous geometries.
problem Solving first BGG operators on specific geometries.
method Approach and find solutions on homogeneous geometries.
result Provide new examples with interesting properties.
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48 results for “homogeneous geometries”
Homogeneous Hypersurfaces in 4-dimensional Thurston Geometries with 4-dimensional Isometry Groupmath.DG
Classifies homogeneous hypersurfaces in specific 4D geometries.
problem Classifying homogeneous hypersurfaces in 4D Thurston geometries.
method Analyzing subalgebras of Lie algebras and isometry groups.
result Determined all homogeneous hypersurfaces up to ambient isometries.
Study BGG operators on homogeneous conformal geometries.
problem Finding solutions to BGG operators on homogeneous conformal geometries.
method Invariant calculus for algebraic computations.
result Explicit solutions found for conformal Killing tensors, forms, and spinors.
Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.
We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed points. We describe the sets of such automorphisms on homogeneous parabolic geometri…
The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.
problem Classifying homogeneous hypersurfaces in specific 4D geometries.
method Analyzing isometry groups and applying classification techniques.
result Homogeneous hypersurfaces identified in , and .
Geodesic orbit and weakly symmetric properties in spray geometry.
problem Understanding properties of homogeneous spray manifolds.
method Using reductive decompositions and spray vector fields to describe and prove properties.
result Weakly symmetric spray manifolds are geodesic orbit manifolds.
Study on homogeneous geodesics in sub-Riemannian geometry.
problem Characterizing and understanding homogeneous geodesics in sub-Riemannian manifolds.
method Criterion for geodesics to be homogeneous, proof of geodesic orbit spaces, examples of geodesic orbit sub-Riemannian manifolds.
result Existence of at least one homogeneous geodesic under broad conditions.
We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact Kähler manifolds.
Classifies polar actions on 3D homogeneous spaces.
problem Classifying polar isometric actions on 3D homogeneous spaces.
method Orbit equivalence classification and study of cohomogeneity one actions.
result Classification of extrinsically homogeneous surfaces and orbit foliations.
We construct a series of examples of non--flat non--homogeneous parabolic geometries that carry a symmetry of the parabolic geometry at each point.
Introduces submersion in spray geometry and defines key components.
problem Defines spray vector field and connection operator in homogeneous spray geometry.
method Introduces submersion technique and uses global invariant frames on Lie groups.
result Proves correspondence between geometric objects in homogeneous spray manifolds.
Classifies homogeneous Riemannian structures on 3D Lie groups.
problem Classifying homogeneous Riemannian structures on 3D Lie groups.
method Classification based on left invariant metrics and previous classifications.
result Complete classification of homogeneous Riemannian structures on 3D Lie groups.
We define a Riemannian structure as a pre-homogeneous geometric structure with curvature R. We show that R=0 if and only if the underlying metric has constant curvature. We define pre-homogeneous geometric structures and pose some problems.
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no referenc…
The paper generalizes cyclic metrics in homogeneous Finsler geometry.
problem Understanding cyclic metrics in homogeneous Finsler spaces.
method Generalization of cyclic metrics, proving conditions for symmetry, and constructing cyclic metrics.
result A Finsler cyclic Lie group with an Abelian Lie algebra.
Study circles to understand dynamics and rigidity in homogeneous spaces.
problem Understanding dynamics and rigidity in infinite-volume homogeneous spaces.
method Addressing four questions about circle packings.
result Highlighting the interplay between dynamics, geometry, and rigidity.
Constructs geometries with nonvanishing curvature and essential automorphisms.
problem Creating geometries with nonvanishing curvature and essential automorphisms.
method Using elements of the kernel of the Kostant Laplacian to construct homogeneous Cartan geometries, then modifying them to make base manifolds compact.
result Infinite families of regular normal Cartan geometries with nonvanishing curvature and essential automorphisms on closed manifolds for higher rank parabolic model geometries.
Geometric study of thermodynamics using cotangent bundles.
problem Formulating classical thermodynamics using geometric structures.
method Contact geometry on cotangent bundles, homogeneous coordinates for intensive variables.
result Geometric formulation of thermodynamics with homogeneity properties.
The paper classifies hypersurfaces in a specific 4D geometry.
problem Classify homogeneous hypersurfaces in the four-dimensional Thurston geometry .
method Used geometric conditions to classify hypersurfaces with constant principal curvatures.
result Complete classification of homogeneous hypersurfaces in .
Develops theory of homogeneous statistical manifolds and classifies Lie groups.
problem Understanding statistical manifolds and Lie groups.
method Constructs examples and classifies Lie groups using information geometry.
result Explicit examples of homogeneous statistical manifolds of low dimension constructed.
Characterizes homogeneous spaces with geometric structures using connections.
problem Characterizing homogeneous spaces with various geometric structures.
method Using connections to characterize reductive homogeneous spaces.
result Generalizes Ambrose-Singer theorem to non-Riemannian geometries.
A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group acting continuously on the geometry, has been obtained by Kramer and Lytchak (Homogeneous compact geometries, Transform. Groups 19 (2016), 43-58 and Err…
Researchers generalize cosmological models using Finsler geometry.
problem Cosmological models that closely approximate pseudo-Riemannian geometry.
method Identifying Lie Algebra of symmetry generators for spatially homogeneous and isotropic Finsler geometries.
result Found the most general spatially homogeneous and isotropic Berwald spacetimes.
Homogeneous Special Geometrymath.DG
Motivated by the physical concept of special geometry two mathematical constructions are studied, which relate real hypersurfaces to tube domains and complex Lagrangean cones respectively. Me\-thods are developed for the classification of homogeneous Riemannian hypersurfaces and for the classification of linear transit…
We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on compact symmetric spaces.
The paper examines the geometry of specific submanifolds in flag manifolds.
problem Understanding the geometry of invariant almost semi Kähler submanifolds.
method Analyzing homogeneous spaces as almost Hermitian submanifolds of flag manifolds.
result Minimal and totally geodesic properties of certain submanifolds.
Study geodesic complexity in homogeneous Riemannian manifolds.
problem Geodesic motion planning and complexity in homogeneous Riemannian manifolds.
method Riemannian geometry, stratifications of cut loci, and properties of homogeneous manifolds.
result Established new bounds on geodesic complexity and computed its value for homogeneous Riemannian manifolds.
We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.
Researchers study solitons on homogeneous spaces, finding useful geometric structures.
problem Finding solitons on homogeneous spaces.
method Investigating solitons with a focus on equivalence relations and optimal tangent directions.
result Solitons on homogeneous spaces have been identified as a useful tool in various geometric contexts.
Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.
problem Understand how algebraic conditions on isotropy group affect the geometry and curvature of Lorentzian homogeneous spaces.
method Prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose--Singer connection with indecomposable, non-irreducible holonomy.
result Generalize existing results about Lorentzian homogeneous spaces with irreducible isotropy and prove results about Lorentzian connections with parallel torsion and 2-symmetric connections.
We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely a…
Limits of geometriesmath.GT
A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…
We show how to specify preferred parameterisations on a homogeneous curve in an arbitrary homogeneous space. We apply these results to limit the natural parameters on distinguished curves in parabolic geometries.
In this survey article we provide an introduction to submanifold geometry in symmetric spaces of noncompact type. We focus on the construction of examples and the classification problems of homogeneous and isoparametric hypersurfaces, polar and hyperpolar actions, and homogeneous CPC submanifolds.
We provide a uniform framework to study the exceptional homogeneous compact geometries of type C3. This framework is then used to show that these are simply connected, answering a question by Kramer and Lytchak, and to calculate the full automorphism groups.
The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.
problem Understanding negatively curved homogeneous Finsler manifolds.
method Generalizing Heintze-Kobayashi-Wolf theory to homogeneous Finsler geometry, proving two main theorems.
result Negatively curved homogeneous Finsler manifolds are isometric to Lie groups with specific properties.
Study path geometries with constant torsion and cone structures.
problem Characterizing path geometries with nontrivial torsion.
method Introducing constant torsion, establishing correspondence with cone structures, describing in terms of integrable systems.
result Path geometries with constant torsion correspond to cone structures on homogeneous ruled surfaces.
Y. Nikonorov completes a proof in a geometry paper.
problem Completing a proof in a geometry paper.
method Completing an argument from a previous proof.
result Proof of Theorem 2.5 in JGA 27 (2017) is now complete.
A new method simplifies contact Hamiltonian mechanics.
problem Traditional contact Hamiltonian mechanics is complex.
method Introduces sections of line bundles over contact manifolds.
result Reduces contact Hamiltonian formalism to symplectic.
In this paper, we explore the similarity between normal homogeneity and -homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected -homogeneous Finsler space is --homo-geneous, for some suitably chosen connected quasi-compact . So -homogeneous Fins…
We investigate geometric properties of homogeneous parabolic geometries with generalized symmetries. We show that they can be reduced to a simpler geometric structures and interpret them explicitly. For specific types of parabolic geometries, we prove that the reductions correspond to known generalizations of symmetric…
The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the…
A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension . In this minicourse we discuss these problems from a ge…
Complete complex parabolic geometries (including projective connections and conformal connections) are flat and homogeneous. This is the first global theorem on parabolic geometries.
Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimen…
We will discuss in this paper homogeneous locally conformally Keahler (or shortly homogeneous l.c.K.) manifolds and locally homogeneous l.c.K. manifolds from various aspects of study in the field of l.c.K. geometry. We will provide a survey of known results along with some new results and observations; in particular we…
Study homogenizes equations on parallelizable manifolds using tensor localization and periodicity.
problem Homogenizing oscillating linear elliptic equations on parallelizable manifolds.
method Two-scale convergence through localization and periodicity induced by geometry.
result Explicit cell formulae for the homogenization limit and a theory of two-scale convergence of tensors.