Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
Reduces Poisson manifolds with Hamiltonian Lie algebroids.
problem Handling Poisson manifolds with Hamiltonian Lie algebroids.
method Introducing compatibility of momentum sections and quotienting zero level sets.
result Quotient space of zero level set of compatible momentum section is a Poisson manifold.
Study connects Lie groups to specific Riemannian manifolds.
problem Understanding Lie groups through Riemannian manifold properties.
method Investigates Lie groups as 3D almost paracontact almost paracomplex Riemannian manifolds.
result Established correspondence between Lie algebra and matrix representation.
Study on 4D Lie groups and related almost hypercomplex manifolds.
problem Characterizing almost hypercomplex manifolds with specific metrics.
method Construction and classification of manifolds based on Lie algebras.
result Established a connection between Lie algebra classes and manifold classifications.
Investigate local Lie group structure of bisections over compact manifolds
problem Study the local Lie group structure associated with the space of admissible bisections of a local Lie groupoid over a compact manifold.
method Investigate the relation of this local Lie group to the Lie algebra of sections of the associated Lie algebroid.
result Prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections.
Study Lie groups as 4D hypercomplex manifolds with specific metrics.
problem Understanding Lie groups with hypercomplex structures in 4D.
method Investigated Lie groups as almost hypercomplex Hermitian-Norden manifolds, established a correspondence between Lie algebras and matrix representations, and constructed examples.
result Explicit matrix representations of Lie groups with hypercomplex structures in 4D.
Directly proves positive injectivity radius for Lie manifolds.
problem Injectivity radius positivity for Lie manifolds.
method Direct, geometric proof.
result Injectivity radius is positive for Lie manifolds.
Integrable LCK manifolds characterized as Kähler Lie algebras.
problem Characterizing LCK manifolds with integrable anti-Lee forms.
method Examining LCK manifolds with integrable anti-Lee forms and applying to Lie algebras.
result Unimodular integrable LCK Lie algebras are Kähler Lie algebras with specific derivations.
The paper classifies Sasaki and Vaisman manifolds of unimodular Lie groups.
problem Understanding the structure of Sasaki and Vaisman manifolds on unimodular Lie groups.
method Basic structure theorem and complete classification for simply connected manifolds.
result A complete classification of simply connected Sasaki and Vaisman unimodular Lie groups.
Study finds specific Lie groups with Kenmotsu structures.
problem Characterizing Lie groups with Kenmotsu structures.
method Determined Lie groups with left invariant Kenmotsu structures.
result These Lie groups are Einstein Riemannian manifolds.
The paper proves a theorem about completely solvable Lie foliations on compact manifolds.
problem Generalizing Haefliger's theorem to completely solvable Lie foliations.
method Proves that every completely solvable Lie foliation on a compact manifold is the inverse image of a homogeneous foliation.
result Every completely solvable Lie foliation on a compact manifold is the inverse image of a homogeneous foliation.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
The paper defines and explores Hom-Lie algebroids and related structures.
problem Defining and studying Hom-Lie algebroids and related algebraic structures.
method Modifying and extending the definitions of Lie algebroids and introducing new structures like Hom-Poisson manifolds, Hom-Lie bialgebroids, and Hom-Courant algebroids.
result Hom-Courant algebroids are shown to have an underlying algebraic structure of Hom-Leibniz algebras or Hom-Lie 2-algebras.
Study Hom-Lie algebroid connections on complex manifolds.
problem Irreducible connections on Hom-Lie algebroids.
method Proved moduli space structure using H-gauge theory.
result Moduli space has a Hausdorff Hilbert manifold structure.
The study finds geodesic orbit metrics on Lie groups from flag manifolds.
problem Investigating geodesic orbit metrics on Lie groups.
method Using generalized flag manifolds to form metrics on simple Lie groups.
result All left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.
The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a com…
Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operator…
New Q-manifolds theory integrates Lie algebroids.
problem Integrating Lie algebroids over smooth manifolds.
method Introducing Q-groupoids and Q-bundles, proving Lie algebroids arise from Q-manifolds.
result Transitive Lie algebroids over second countable, smooth manifolds are integrated to locally trivial Q-groupoids.
Introduces comomentum sections and proves they are Poisson maps.
problem Generalizing Poisson maps to Hamiltonian Lie algebroids.
method Introduces comomentum sections and proves they are Lie algebroid morphisms and Poisson maps.
result Comomentum sections are Poisson maps between proper Poisson manifolds.
Compact Lie groups G allow topological G-manifolds to be approximated by finite G-CW complexes.
problem Understanding the homotopy type of topological G-manifolds for compact Lie groups G. method Using countable G-CW complexes to approximate the G-homotopy type of topological G-manifolds. result Topological G-manifolds have the G-homotopy type of finite-dimensional countable G-CW complexes. Abstract reviews Markov processes with jumps on manifolds and Lie groups.
problem Analyzing Markov processes with jumps in geometric settings.
method Stochastic differential equations, Courrège theorem, invariant Markov processes.
result Developments in Lie groups and manifolds under various actions.
Study Einstein Lie groups and geodesic orbit manifolds, finding some are not geodesic orbit.
problem Characterizing Einstein Lie groups and geodesic orbit manifolds.
method Characterizing GimesK-invariant geodesic orbit metrics on Lie groups G for regular subgroups K. result Extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds.
The paper characterizes flat affine connections on manifolds and Lie groups.
problem Characterizing flat affine connections on manifolds and Lie groups.
method New characterization through affine representations of automorphisms.
result Existence of a Lie group with a flat affine bi-invariant connection.
Study of unimodular Sasaki and Vaisman Lie groups, determining all modifications explicitly.
problem Classifying unimodular Sasaki and Vaisman Lie groups.
method Applying the technique of modification to determine all homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups explicitly.
result Complete classification of unimodular Sasaki and Vaisman Lie groups.
Develops a spectral sequence for Lie group actions on manifolds.
problem Understanding cohomology of manifolds with Lie group actions.
method Introduces a spectral sequence relating manifold cohomology to Lie algebra cohomology.
result Establishes a new description of de Rham cohomology for manifolds with Lie group actions.
Extends Cheeger's method to Lie groupoid actions on manifolds.
problem Smooth Lie group actions on manifolds with singularities.
method Extension of Cheeger's deformation techniques to Lie groupoid actions.
result Explicit sectional curvature description of the deformation.
New Lie systems defined on k-contact manifolds, with applications.
problem Understanding Lie systems on k-contact manifolds. method Distributional approach to k-contact manifolds and Hamiltonian vector fields. result Lie systems can be understood as Hamiltonian relative to a k-contact manifold. Examines Lie algebras for symmetries in odd-dimensional manifolds.
problem Symmetry analysis in odd-dimensional manifolds.
method Analyzes local Lie algebras of pairs of functions.
result Identifies infinitesimal symmetries of almost-cosymplectic-contact structures.
We define Lie and Courant algebroids on Fréchet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fréchet manifold and show that it inherits a Fréchet Lie algebroid structure. We show that the Lie algebroid cohomology of the $\bb$-cotangent bundle Lie algebroid of a weakly sympl…
Defines and extends Lie algebroid prolongations in convenient settings.
problem Adapting Lie algebroid prolongations to convenient settings.
method Defined and adapted Lie algebroid prolongations over fibred manifolds.
result Stability of prolongations under projective and direct limits.
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.
problem Generalizing Riemann-Roch theorem for manifolds with regular foliations.
method Developed Lie algebroid index theory and applied it to obtain a generalized Riemann-Roch theorem.
result Obtained a generalized Riemann-Roch theorem for manifolds with regular foliations.
Generalizes Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms.
problem Understanding the algebraic structure of conformal Killing-Yano forms.
method Proposes a new Lie bracket and shows graded Lie algebra properties in constant and Einstein manifolds.
result Conformal Killing-Yano forms form a graded Lie algebra in specific manifolds.
Lie PCA improves density estimation on symmetric manifolds.
problem Density estimation for symmetric manifolds.
method Spectral method to approximate Lie algebra of symmetry group.
result Improved sample complexity and density estimation on various data sets.
Study weightings from singular Lie filtrations.
problem Generalize constructions for singular Lie filtrations.
method Study weightings arising from singular Lie filtrations.
result Generalizes constructions for (regular) Lie filtrations.
Study of 3D Lie group structures with special Riemannian properties.
problem Understanding curvature properties of specific Lie group manifolds.
method Construct and analyze almost paracontact almost paracomplex Riemannian manifolds on Lie groups.
result Curvature properties of constructed manifolds on Lie groups are investigated.
A manifold with a ``Lie structure at infinity'' is a non-compact manifold M0 whose geometry is described by a compactification to a manifold with corners M and a Lie algebra of vector fields on M, subject to constraints only on M∖M0. The Lie structure at infinity on M0 determines a metric on $M_…
New Lie groups found for Poisson diffeomorphisms.
problem Finding Lie group structures on Poisson diffeomorphism groups.
method Using Poisson groupoids, develop Lie group structures.
result Poisson diffeomorphism groups of various Poisson manifolds are regular Lie groups.
Introduces modular class for Lie algebroids with Nambu structures.
problem Modular classes of Nambu-Poisson manifolds.
method Definition of modular class for Lie algebroids with Nambu structures and properties.
result Properties of modular classes extend to Lie algebroids with Nambu structures.
Develops Lie algebraic approach for compact complex homogeneous manifolds.
problem Proves important results on compact complex homogeneous manifolds.
method Uses standard results in Lie theory to associate a canonical abelian Lie algebra with a given integrable complex structure.
result Provides a new method of associating a canonical abelian Lie algebra with a given integrable complex structure.
Discuss various definitions of vector fields on manifolds leading to Lie algebras.
problem Defining vector fields on convenient manifolds.
method Various definitions of vector fields and their equivalence.
result Equivalent definitions of vector fields on manifolds.
Study examines Lie algebroids with homological sections, generalizing Q-manifolds and Lie superalgebras.
problem Exploring Lie algebroids with homological sections.
method Derived bracket formalism to define an odd Loday-Leibniz bracket on sections.
result Sections of inner Q-algebroids come equipped with an odd Loday-Leibniz bracket.
The paper introduces a new form on Lie algebroids over multisymplectic manifolds.
problem Higher generalizations of Poisson structures and momentum maps.
method Introducing a compatible E-n-form on Lie algebroids.
result The introduced form satisfies a compatibility condition with Lie algebroid and multisymplectic structures.
Characterizes connections on normal distributions manifold.
problem Geometric characterization of connections on normal distributions.
method Homogeneous statistical manifold structure and Lie group analysis.
result Geometric characterization of α-connections on Lie group. We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which ``controls'' deformations of the structure bracket of the algebroid. We also have a closer look at various special cases such as Lie algebras, Poisson manifolds, foliations, Lie algebra actions on manifo…
We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional Z-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like TTM, e…
The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some kno…
The paper classifies manifolds acted upon by a specific Lie group.
problem Characterizing manifolds acted upon by a specific Lie group.
method Analyzing the structure of manifolds under isometric action of a Lie group.
result Characterization of the structure of manifolds under specific Lie group action.