The notion of a complex hyperpolar action on a symmetric space of non-compact type has recently been introduced as counterpart of a hyperpolar action on a symmetric space of compact type. In this paper, we construct examples of a complex hyperpolar action without singular orbit and investigate the geometry of the orbit…
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48 results for “hyperpolar actions”
We study hyperpolar actions on reducible symmetric spaces of the compact type. Our main result is that an indecomposable hyperpolar action on a symmetric space of the compact type is orbit equivalent to a Hermann action or of cohomogeneity one.
The study classifies and investigates translators invariant under hyperpolar actions on symmetric spaces.
problem Understanding translators invariant under hyperpolar actions on symmetric spaces.
method Classification and investigation of translators given by functions invariant under hyperpolar actions.
result Classification and investigation of translators in symmetric spaces under hyperpolar actions.
We show that polar actions of cohomogeneity two on simple compact Lie groups of higher rank, endowed with a biinvariant Riemannian metric, are hyperpolar. Combining this with a recent result of the second-named author, we are able to prove that polar actions induced by reductive algebraic subgroups in the isometry grou…
Path group actions induced by sigma-actions and affine Kac-Moody symmetric spaces of group typemath.DG
Extends PF submanifold results and connects Kac-Moody spaces.
problem Computational results and submanifold geometry of PF actions.
method Defines isomorphism between Hilbert spaces, shows equivariance, and uses parallel transport.
result Shows natural isomorphism between Kac-Moody spaces of group type.
We show that the equivariant cohomology of any hyperpolar action of a compact and connected Lie group on a symmetric space of compact type is a Cohen-Macaulay ring. This generalizes some results previously obtained by the authors.
Reduces proper actions to simpler core actions for analysis.
problem Understanding properties of proper actions on manifolds.
method Extending Skjelbred and Straume's construction to non-compact groups, focusing on core of actions.
result Properties of proper actions are determined by simpler core actions.
We prove that an isometric action of a compact Lie group on a compact symmetric space is variationally complete if and only if it is hyperpolar.
This paper has been withdrawn by the author due to a crucial error related with the consideration of regular points
We study isometric actions on Riemannian symmetric spaces of noncompact type which are induced by reductive algebraic subgroups of the isometry group. We show that for such an action there exists a corresponding isometric action on a dual compact symmetric space, which reflects many properties of the original action. F…
We study isometric Lie group actions on symmetric spaces admitting a section, i.e. a submanifold which meets all orbits orthogonally at every intersection point. We classify such actions on the compact symmetric spaces with simple isometry group and rank greater than one. In particular we show that these actions are hy…
A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersect…
We study polar actions with horizontal sections on the total space of certain principal bundles with base a symmetric space of compact type. We classify such actions up to orbit equivalence in many cases. In particular, we exhibit examples of hyperpolar actions with cohomogeneity greater than one on locall…
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.
A foliation on a Riemannian manifold is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold that intersects each leaf of the foliation orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space of noncompact type.
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.
Study on actions on symmetric spaces, focusing on orbit properties.
problem Investigating properties of orbits in symmetric spaces related to .
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 .
Classical and quantum Hamiltonian reductions of free geodesic systems of complete Riemannian manifolds are investigated. The reduced systems are described under the assumption that the underlying compact symmetry group acts in a polar manner in the sense that there exist regularly embedded, closed, connected submanifol…
The paper proves a Calabi-Yau structure on complexifications of rank two symmetric spaces.
problem Existence of Calabi-Yau structures on complexifications of symmetric spaces.
method Using orbit geometry and shape operators, the authors prove the existence of a Calabi-Yau structure.
result A Calabi-Yau structure exists on the complexification of rank two symmetric spaces.
The paper introduces austere and arid submanifolds in Hilbert spaces.
problem Classifying minimal orbits in hyperpolar PF actions on Hilbert spaces.
method Introducing austere and arid submanifolds into PF submanifolds in Hilbert spaces.
result Examples of infinite dimensional austere and arid PF submanifolds in Hilbert spaces.
The paper studies curvatures and austere properties of orbits in symmetric spaces.
problem Analyzing curvatures and austere properties of orbits in symmetric spaces.
method Using Hermann actions and hyperpolar properties, the paper derives explicit formulas for principal curvatures and conditions for orbits to be austere.
result The paper provides conditions for orbits to be austere and extends previous results to a larger class of infinite-dimensional submanifolds.
In this paper, we assume that all isoparametric submanifolds have flat section. The main purpose of this paper is to prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type admits a reflective focal submanifold and if it of real analy…
Topics in polar actionsmath.DG
Lectures on polar actions and their properties in Riemannian geometry.
problem Characterizing polar actions and understanding their properties.
method Analyzing isometric actions on Riemannian manifolds, using normal slice theorem and principal orbit type theorem.
result Characterization of polar actions in terms of integrability of the distribution of normal spaces to the principal orbits.
We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.
Classifies polar foliations on symmetric spaces.
problem Classifying polar foliations on symmetric spaces.
method Orbit equivalence and classification up to codimension two.
result Foliations are either hyperpolar or extensions of rank one foliations.
Homologically trivial symplectic cyclic actions need not extend to Hamiltonian circle actionsmath.SG
We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to stu…
Introduces Conditional Action Trees to simplify RL action spaces.
problem Challenges in RL with large, complex action spaces.
method Structures action spaces and reduces complexity through Conditional Action Trees.
result Demonstrates effectiveness in reducing action space and improving decision making.
Macro Action Reinforcement Learning with Sequence Disentanglement using Variational Autoencodercs.LG
One problem in the application of reinforcement learning to real-world problems is the curse of dimensionality on the action space. Macro actions, a sequence of primitive actions, have been studied to diminish the dimensionality of the action space with regard to the time axis. However, previous studies relied on human…
Study polar actions on Damek-Ricci spaces, proving existence and finding examples.
problem Characterize polar actions on Damek-Ricci spaces.
method Prove criteria for isometric actions to be polar, find examples, and classify actions.
result Non-trivial polar actions exist on all Damek-Ricci spaces.
New RL algorithm tackles complex discrete action spaces.
problem Challenges in applying on-policy RL in high-dimensional discrete action spaces.
method Action-value critic, correlated actions, gradient sparsification.
result Empirically outperforms related on-policy algorithms.
Reduction principles for proper actions on smooth manifolds.
problem Proper actions on smooth manifolds and their properties.
method Exhibit constructions and prove reduction principles for proper actions.
result Reduction principles hold for proper actions, polar actions, and copolarity.
Sections of polar actionsmath.DG
Totally geodesic sections found in polar actions.
problem Understanding sections of polar actions on Riemannian manifolds.
method Elementary proof of a folklore result.
result Sections of polar actions are totally geodesic.
Simplifies large action space bandits by selecting representative actions.
problem Efficiently managing large action spaces with correlated outcomes.
method Random sampling and solving of bandit instances to identify representative actions.
result The algorithm selects a smaller set of representative actions that perform nearly as well as the full action space.
Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
Conditions for reducing quasi-actions to tree actions and group properties.
problem Conditions for reducing quasi-actions to tree actions.
method Reduction to cobounded isometric actions on trees.
result Groups with quasi-orbits quasi-isometric to trees are virtually free.
The study proves conditions for symplectic torus actions on manifolds with non-contractible orbits.
problem Conditions for symplectic torus actions with non-contractible orbits.
method Analyzes symplectic torus actions on manifolds, proving conditions for Hamiltonian actions and orbit properties.
result Symplectic actions with non-contractible orbits are not Hamiltonian unless the orbits are contractible.
We identify action representations from video data, proving their statistical benefits.
problem Identifying latent action policies from video data.
method Entropy-regularized LAPO objective, formalizing desiderata for action representations.
result Entropy-regularized LAPO identifies action representations satisfying desiderata under suitable conditions.
Study properties of orbits of Hermann actions without commutability assumptions.
problem Investigate geometric properties of orbits of Hermann actions.
method Compute the second fundamental form and provide conditions for weak reflection and aridity.
result Sufficient conditions for weak reflection and aridity of orbits of Hermann action.
We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.
A new method learns action representations for reinforcement learning.
problem Efficient action-value estimation in reinforcement learning.
method Action hypergraph networks framework for learning action representations.
result Hypergraph Q-networks show effectiveness on various domains.
New reinforcement learning framework for adapting to new actions.
problem Making reinforcement learning agents adaptable to new actions without retraining.
method Two-stage framework: infer action representations first, then train a flexible policy.
result Agents can make decisions from new action sets without retraining.
AQL uses amortized inference to handle high-dimensional action spaces in Q-learning.
problem Difficulty in maximizing over large action spaces in Q-learning.
method Replace expensive maximization over all actions with a maximization over a small subset sampled from a learned proposal distribution.
result AQL outperforms existing methods on continuous control tasks with up to 21 dimensional actions.
Defines and computes a generalized spectral action for Lorentz warped products.
problem Computing spectral actions for Lorentz warped products.
method Defines and computes the bimetric spectral Einstein-Hilbert action for Lorentz warped products.
result Derives a Kastler-Kalau-Walze type theorem for Lorentz warped products.
The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.
problem Exploring acylindrical actions on trees and their properties.
method Demonstrates criteria for preserving acylindrical hyperbolicity and analyzes the outer automorphism group of Baumsligar-Solitar groups.
result Proves acylindrical hyperbolicity of non-solvable Baumsligar-Solitar groups.
The study examines largest hyperbolic actions in groups and finds many do not exist.
problem Identifying largest hyperbolic actions in groups.
method Analysis of equivalence classes of cobounded actions on hyperbolic metric spaces.
result Many families of groups, including 3-manifold groups and mapping class groups, do not have largest hyperbolic actions.
UTE improves reinforcement learning by measuring action uncertainty, enhancing policy learning efficiency.
problem Degrading performance of action repetition in reinforcement learning, especially with sub-optimal actions.
method UTE uses ensemble methods to measure uncertainty during action extension, allowing strategic exploration or certainty.
result UTE outperforms existing action repetition algorithms, significantly enhancing policy learning efficiency.
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.
Proper actions on bornological spaces are characterized with compatible coarse structures.
problem Characterizing proper actions on bornological spaces.
method Proving the existence of compatible coarse structures for proper actions.
result Bornological spaces admit compatible coarse structures for proper actions.