Smooth approximations for continuous functions on orbit spaces.
problem Approximating continuous functions on orbit spaces.
method Study of subcartesian spaces and proper Lie group actions.
result Continuous functions can be approximated by smooth functions.
Study classifies compact geodesic orbit spaces with two isotropy components.
problem Characterizing geodesic orbit Riemannian spaces.
method Classification of spaces with specific isotropy properties.
result Classification of compact geodesic orbit spaces with two isotropy summands.
The classification of G-spaces by Palais is refined for the case where the orbit space satisfies certain mild topological hypotheses. It is shown that when a sequence of such orbit spaces is "close" to a limit orbit space, in some suitable sense, within a larger ambient orbit space, the G-spaces in the tail of the sequ…
Study orbit spaces of equivariant ANEs for proper actions of metrizable groups.
problem Understanding the extension properties of orbit spaces for proper actions.
method Analyzing equivariant absolute neighborhood extensors for proper G-spaces. result Proving conditions under which orbit spaces of metrizable G-orbits are ANEs. The study characterizes geodesic orbit Riemannian spaces and their properties.
problem Characterizing geodesic orbit Riemannian spaces and their properties.
method Analyzing the structure of nilradical and radical of Lie algebra, discussing compact Lie group representations.
result Described the structure of nilradical and radical of Lie algebra of isometry group.
Circle actions yield different orbit spaces.
problem Understanding differential structures on orbit spaces.
method Analyzing non-isomorphic linear circle actions.
result Non-diffeomorphic orbit spaces from different actions.
New Frobenius manifold structures found on Dicyclic group orbits.
problem Finding Frobenius manifold structures on orbits spaces of Dicyclic groups.
method Applying Dubrovin's method to Dicyclic groups.
result Dicyclic group orbits spaces acquire two Frobenius manifold structures.
Researchers classify geodesic orbit spaces for compact Lie groups of rank two.
problem Identifying geodesic orbit spaces for compact Lie groups of specific rank.
method Classification of simply connected geodesic orbit spaces where G is a compact Lie group of rank two.
result Only certain spheres and projective spaces, with metrics induced from Hopf fibrations, are geodesic orbit spaces for compact Lie groups of rank two.
Classifies actions on Minkowski space up to orbit equivalence.
problem Classifying actions on Minkowski space up to orbit equivalence.
method Classifying actions up to orbit equivalence, providing representations and orbit spaces.
result Orbits and orbit spaces determined for proper actions.
Geodesic orbit metrics proven on specific homogeneous spaces.
problem Geodesic orbit metrics on homogeneous spaces.
method Using strongly isotropy irreducible spaces and proving natural reductivity.
result Geodesic orbit metrics are naturally reductive on constructed homogeneous spaces.
Study geodesic orbit metrics in quaternionic Stiefel manifolds.
problem Characterize geodesic orbit spaces in quaternionic Stiefel manifolds.
method Analyze homogeneous Riemannian spaces (M=G/H,g) with geodesics as orbits of subgroups. result Identify conditions for $(\Sp(n)/\Sp(n_1) imes \cdots imes \Sp(n_s), g)$ to be a geodesic orbit space.
Classifies geodesic orbit spaces with abelian isotropy subgroups.
problem Characterizing and classifying geodesic orbit spaces with specific isotropy subgroups.
method Simplified study of geodesic orbit metrics on G/S by reducing to submanifolds and generalized flag manifolds, using properties of root systems.
result Geodesic orbit spaces of the form (G/S,g) are naturally reductive.
The study counts periodic orbits on smooth manifolds, adding ghost orbits for completeness.
problem Counting periodic orbits of vector fields on smooth closed manifolds.
method Enlarging the space of orbits to include ghost orbits, defining weight functions, and showing constancy under deformation.
result The weight function remains constant as the vector field moves and Γ deforms. The paper classifies geodesic orbit spaces with simple isotropy groups.
problem Classifying geodesic orbit spaces with simple isotropy groups.
method Classifying G-naturally reductive and G-geodesic orbit metrics on M. result Classification of geodesic orbit spaces with simple isotropy groups.
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
Study of differential forms and vector fields on orbit spaces.
problem Understanding vector fields and differential forms on orbit spaces.
method Defined differential forms and vector fields as multilinear maps on infinitesimal diffeomorphisms.
result Intrinsic view of vector fields and differential forms on orbit spaces.
We study isometric cohomogeneity one actions on the (n+1)-dimensional Minkowski space up to orbit-equivalence. We give examples of isometric cohomogeneity one actions on the Minkowski space whose orbit spaces are non-Hausdorff. We show that there exist isometric cohomogeneity one actions on the Minkowski space which ar…
New measures on orbit spaces for orthogonal groups identified.
problem Characterizing measures on orbit spaces of orthogonal groups.
method Constructing Hilbert measures on orbit spaces of coregular representations of orthogonal groups.
result Hilbert measures have singularities if and only if the number of copies equals the dimension.
Criterion for periodic orbits convergence proved.
problem Periodic orbits convergence criterion.
method Criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups.
result Criterion for periodic orbits convergence proved.
We list up all the possible local orbit types of hyperbolic or elliptic orbits for the isotropy representations of semisimple pseudo-Riemannian symmetric spaces. It is key to give a recipe to determine the local orbit types of hyperbolic principal orbits by using three kind of restricted root systems and Satake diagram…
The abstract discusses extensions of Jacobi groups and their orbit space properties.
problem Understanding the properties of extended Jacobi groups and their orbit spaces.
method Proving an analogue of Chevalley Theorem and constructing a Frobenius structure.
result Construction of a Dubrovin Frobenius structure on the orbit space.
Reconstruct flows from their orbit spaces using group actions.
problem Reconstructing flows from their orbit spaces.
method Using group actions and pseudo-Anosov flows.
result Reconstruct flows from their orbit spaces.
Differentiable spaces derived from Lie group actions have vector fields and forms.
problem Understanding the differential structure of orbit spaces of Lie group actions.
method Analyzing the differential structure of orbit spaces of proper Lie group actions on smooth manifolds.
result Orbit spaces of Lie group actions are differentiable spaces with exterior algebra of differential forms.
In this paper we prove that an isometry between orbit spaces of two proper isometric actions is smooth if it preserves the codimension of the orbits or if the orbit spaces have no boundary. In other words, we generalize Myers-Steenrod's theorem for orbit spaces. These results are proved in the more general context of s…
Lifts isometries in orbit spaces for compact groups.
problem Isometries in orbit spaces of compact groups.
method Equivariant isometry of original Euclidean space.
result Simple formula for connected component of isometry group.
Study geodesic orbit metrics on specific homogeneous spaces.
problem Characterize geodesic orbit spaces in a class of homogeneous bundles.
method Analyze geodesic orbit spaces of compact Lie groups with semisimple subgroups.
result Identify conditions for a metric to be geodesic orbit.
No exceptional orbits found in Hilbert spaces actions.
problem Proving the non-existence of exceptional orbits in Hilbert spaces.
method Analyzing polar actions on separable Hilbert spaces by connected Lie groups.
result Proved non-existence of exceptional orbits in Hilbert spaces.
The paper finds conditions for biharmonic orbits in symmetric spaces.
problem Conditions for biharmonic orbits in symmetric spaces.
method Analyzes isotropy representations and biharmonic submanifolds in hyperspheres.
result Necessary and sufficient conditions for biharmonic orbits in symmetric spaces.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
New manifold structures on Weyl group orbit spaces proven.
problem Constructing generalized Frobenius manifold structures.
method Construction on orbit spaces of affine Weyl groups.
result Monodromy groups are parabolic subgroups.
Study classifies Lie group representations with non-empty boundary orbit space.
problem Classifying representations of Lie groups with non-empty boundary orbit space.
method Detailed calculations based on previous work.
result Classification of Lie group representations with non-empty boundary orbit space.
Study on orbit spaces with curvature bounds.
problem Understanding the curvature bounds of orbit spaces.
method Analyzing isometric actions on unit spheres.
result The infimum of sectional curvatures is 1 for most actions.
The paper studies geodesic orbit Finsler metrics on Euclidean spaces.
problem Understanding Finsler metrics on Euclidean spaces with specific geometric properties.
method Analyzing Finsler metrics on Euclidean spaces to show they are fiber bundles over symmetric spaces.
result Finsler metrics on Euclidean spaces are fiber bundles over symmetric spaces with specific geometric properties.
The study confirms a conjecture about polynomials related to symmetric spaces.
problem Understanding polynomials associated with isotropy orbits of symmetric spaces.
method Identified Reiswich's polynomials as special cases of Jacobi polynomials and proved their conjecture.
result The polynomials have pairwise different real roots in the interval [0,1].
Geodesic orbit property studied for Lorentz manifolds.
problem Geodesic orbit property for Lorentz manifolds.
method Defined naturally reductive for pseudo-Riemannian manifolds and proved theorems for Lorentz nilmanifolds.
result Geodesic orbit property holds for Lorentz nilmanifolds under specific conditions.
Researchers find Frobenius manifold structures on orbits spaces of finite groups.
problem Understanding Frobenius manifold structures on orbits spaces of finite groups.
method Applying Dubrovin's method to various orbits spaces of linear representations of finite groups.
result Discoveries of non-trivial Frobenius manifold structures on orbits spaces.
Study finds homogeneous spaces with geodesic orbits but no integrable distributions.
problem Characterizing homogeneous spaces with specific geometric properties.
method Examined Lie groups with compact stabilizers and classified spaces based on geodesic orbits and integrable distributions.
result Identified homogeneous spaces with geodesic orbits but lacking integrable invariant distributions.
The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
problem Classifying Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
method Examining orbits as CR submanifolds, proving total geodesy, and analyzing contact structures.
result Completely determine Sasaki-Einstein orbits.
A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is general…
Geodesic graphs for special Finsler metrics on spheres are studied.
problem Characterizing geodesic orbit Finsler metrics on spheres.
method Explicit constructions and group extensions.
result Not all projective spaces admit invariant Finsler metrics.
The paper studies mean curvature flows on specific orbits of Hermann actions.
problem Analyzing mean curvature flows on principal orbits of Hermann actions.
method Using Mathematica to illustrate and calculate the flows and orbits.
result The minimal principal orbit's position is calculated and illustrated.
We show that, if the family \cal{O} of orbits of all vector fields on a subcartesian space P is locally finite and each orbit in \cal{O} is locally closed, then \cal{O} defines a smooth Whitney A stratification of P. We also show that the stratification by orbit type of the space M/G of orbits of a proper action of a L…
Lectures explore how differential methods improve understanding of algebraic group orbit spaces.
problem Understanding structure of invariants and orbit spaces of algebraic Lie groups.
method Combines algebraic and differential viewpoints to study orbit spaces.
result Differential approach provides deeper insights into invariants and orbit spaces.
This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant Kählerian structure. These coadjoint orbits are defined for a real semi-simple connected non-compact Lie group G w…
The paper solves orbital integrals on Lorentzian symmetric spaces.
problem Determining a function in terms of its orbital integrals on Lorentzian symmetric spaces.
method Extending Helgason's and Orloff's methods to odd-dimensional isotropic Lorentzian symmetric spaces and studying orbital integrals on solvable ones.
result An inversion formula for orbital integrals on solvable Lorentzian symmetric spaces.
Classifies finite orbits of mapping class group actions on surface group representations.
problem Finite orbits of mapping class group actions on surface group representations.
method Classification based on surface genus and representation properties.
result Finite orbits correspond to homomorphisms with finite image for genus at least two, and to finite or special dihedral representations for genus one.
Characterizes pseudo-Anosov orbit spaces via bifoliated planes
problem Characterizing actions on bifoliated planes arising from pseudo-Anosov flows
method Using branched covers, veering triangulations, and a compactness criterion
result Extends previous work on the special case with no odd-prong singularities
Study on G2 actions on symmetric spaces, focusing on orbit properties.
problem Investigating properties of orbits in symmetric spaces related to G2. method Classification and analysis of orbits as Riemannian submanifolds, focusing on principal curvatures and specific types of orbits.
result Classification and properties of orbits in symmetric spaces related to G2.