Symplectic resolves orbifolds with uniform isotropy.
problem Symplectic resolution of orbifolds with homogeneous isotropy.
method Constructing symplectic resolutions for orbifolds with specific isotropy properties.
result Symplectic resolutions achieved for orbifolds with uniform isotropy.
Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie algebras and we classify uniform Lie algebras with five or fewer generators. We defi…
New method predicts nonfactuality in LLM responses using semantic isotropy.
problem Assessing trustworthiness of long-form LLM responses efficiently.
method Semantic isotropy of text embeddings on the unit sphere.
result Higher semantic isotropy correlates with lower factual consistency.
The paper tackles isotropy of symplectic forms using Hodge flows.
problem Whether symplectic forms in a given class are isotropic.
method Introduces nonlinear Hodge heat flows to study isotropy.
result The flow converges to the symplectic form ω smoothly for any initial symplectic form in the class. The study classifies Riemannian homogeneous spaces with polar isotropy actions.
problem Characterizing Riemannian homogeneous spaces with polar isotropy actions.
method Analyzing simply connected Riemannian homogeneous spaces of compact semisimple Lie groups and various non-compact spaces.
result Classification and non-polar isotropy actions for specific spaces.
Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spac…
The paper introduces isotropy as a regularizer to enhance portfolio stability.
problem Model uncertainty and estimation errors in diversification strategies.
method Integrates isotropy as a geometric regularizer into mean-variance optimization.
result Isotropy constraint systematically induces negative average-signal exposure, providing a robust crash hedge.
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].
For Γ a relatively hyperbolic group, we construct a model for the universal space among Γ-spaces with isotropy on the family VC of virtually cyclic subgroups of Γ. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in $O^+(n,1)= \iso(\mathbb H^…
Study isotropy groups for complex orthogonal and skew-symmetric matrices.
problem Understanding isotropy subgroups of orthogonal similarity transformations.
method Analysis of group structure of nonsingular block matrices.
result Group structure of isotropy subgroups related to block Toeplitz matrices.
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.
Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.
problem Computing isotropy subgroups of orthogonal similarity on symmetric matrices.
method Algorithmic procedure solving a Toeplitz matrix equation.
result Structure of isotropy subgroups described.
Classifies rational differential forms on the Riemann sphere based on their isotropy group.
problem Classifying rational differential forms on the Riemann sphere based on their isotropy group.
method Analyzing the isotropy group of rational differential forms with simple poles and zeros.
result Finite subgroups of PSL(2, C) are realizable as isotropy groups for rational 1-forms on the Riemann sphere.
Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
problem Computing isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
method Algorithm for solving a matrix equation to compute isotropy subgroups.
result Computed isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
Study the smallest Laplace eigenvalue in special geometric spaces.
problem Finding the smallest positive eigenvalue of Laplace-Beltrami operator in strongly isotropy irreducible spaces.
method Explicit expression for simply connected cases, proving Einstein manifold properties and eigenvalue bounds.
result Proved E<λ1≤16E for all strongly isotropy irreducible spaces. 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.
We classify all simply connected Riemannian manifolds whose isotropy groups act with cohomogeneity less than or equal to two.
Given a singular foliation, we attach an "essential isotropy" group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with t…
The space of G-invariant metrics on a homogeneous space G/H is in one-to-one correspondence with the set of inner products on the tangent space $\fr{m}\cong T_{\it o}(G/H)$, which are invariant under the isotropy representation. When all the isotropy summands are inequivalent to each other, then the metric is calle…
The concept of an objective spatial direction in special relativity is investigated and theories assuming light-speed isotropy while accepting the existence of a privileged spatial direction are classified. A natural generalization of the proper time principle is introduced which makes it possible to devise experimenta…
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.
New spaces identified with specific properties.
problem Characterizing homogeneous spaces with quaternionic structures.
method Analyzing pseudo-Riemannian almost quaternionic homogeneous spaces with irreducible isotropy.
result Spaces are locally isometric to quaternionic Kähler symmetric spaces under certain conditions.
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.
Study equigeodesics on compact homogeneous spaces using Lie algebra properties.
problem Identifying equigeodesic vectors on compact homogeneous spaces.
method Formula for equigeodesic vectors based on isotropy representation and Lie algebra structure.
result Identification of equigeodesic vectors solely through Lie algebra properties.
We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…
The paper classifies invariant generalized complex structures on specific flag manifolds.
problem Classifying invariant generalized complex structures on partial flag manifolds.
method Proved that invariant generalized almost complex structures are constant in each component of the isotropy representation.
result All invariant generalized complex structures on partial flag manifolds with at most four isotropy summands are classified.
We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 2 of 3) classifies those in which the linear isotropy representation is either irreducible or trivial. The 5-dimensional geometries with irreducible isotropy are the irreducible Riemannian symmetric spaces, while …
The isotropy action on certain symmetric spaces is shown to be equivariantly formal.
problem Understanding the equivariant formality of isotropy actions on symmetric spaces.
method Developed a new approach to prove equivariant formality for (Z2⊕Z2)-symmetric spaces. result Symmetric spaces with (Z2⊕Z2)-symmetry are equivariantly formal and formal in the Sullivan sense. 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.
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.
New classification for curved manifolds with specific symmetries.
problem Classifying non-negatively curved manifolds with specific symmetries.
method Extending equivariant classification results for manifolds with isotropy-maximal or strictly almost isotropy-maximal torus actions.
result Almost isotropy-maximal manifolds are diffeomorphic to products of spheres.
We classify homogeneous pseudo-Riemannian manifolds of index 4 which admit an invariant almost hyper-Hermitian structure and an H-irreducible isotropy group. The main result is that all these spaces are flat except in dimension 12.
Study on invariant Einstein metrics on specific flag manifolds.
problem Existence of invariant Einstein metrics on real flag manifolds.
method Analysis of isotropy representations and Riemannian metrics.
result Existence of non-diagonal Einstein metrics on real flag manifolds.
We obtain an algorithmic construction of the isotropy lattice for a lifted action of a Lie group G on TM and T∗M based only on the knowledge of G and its action on M. Some applications to symplectic geometry are also shown.
We use the equivariant Yang-Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4-manifolds.
New method finds open subsets with trivial holonomy for certain geometries.
problem Finding open subsets with trivial holonomy for Cartan geometries.
method Analyzing the behavior of isotropies in model geometries to generalize properties of isolated higher-order fixed points.
result Existence of open subsets with trivial holonomy for Cartan geometries with certain isotropies.
We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homo…
New method for linearly determining Lie groups from data.
problem Linear determination of Lie groups from data.
method Polar representation and isotropy subgroup data.
result Linear determination possible in general reducible case.
The paper finds multiple Einstein metrics on Stiefel manifolds.
problem Invariant Einstein metrics on Stiefel manifolds with specific isotropy summands.
method Viewed as a total space over generalized flag manifolds, the paper proves the existence of at least four invariant Einstein metrics.
result At least four invariant Einstein metrics are found, including two new ones.
This paper classifies equivariant principal bundles over a 2-sphere using isotropy representations.
problem Classifying equivariant principal bundles over the 2-sphere.
method Using isotropy representations to classify bundles over the 2-sphere.
result Equivariant principal bundles over the 2-sphere can be classified by a Γ-fixed set of homotopy classes of maps and first Chern class.
We introduce a new construction, the isotropy groupoid, to organize the orbit data for split Γ-spaces. We show that equivariant principal G-bundles over split Γ-CW complexes X can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex $A=Γ\backsl…
We show that the isotropy action of a homogeneous space G/K, where G and K are compact, connected Lie groups and K is defined by an automorphism on G, is equivariantly formal and that (G,K) is a Cartan pair.
The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.
problem Classifying flag manifolds with specific isotropy components and finding conditions for Kähler-like scalar curvature.
method Investigating invariant almost Hermitian structures on generalized flag manifolds with two or three irreducible components.
result Classification of flag manifolds admitting Kähler-like scalar curvature and conditions for such structures.
Ricci flow on certain homogeneous spaces creates metrics with positive curvature.
problem Finding metrics with positive Ricci curvature on specific homogeneous spaces.
method Normalized Ricci flow on simply connected homogeneous spaces with two equivalent isotropy summands.
result Every G-invariant metric evolves to one with positive Ricci curvature under Ricci flow.
We show that Jacobi fields along harmonic maps between suitable spaces preserve conformality, holomorphicity, real isotropy and complex isotropy to first order; this last being one of the key tools in the proof by Lemaire and the author of integrability of Jacobi fields along harmonic maps from the 2-sphere to the comp…
We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an…
Let M0n be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M∈M0n, then M is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres…
A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution stru…