Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.
problem Existence and integration of elliptic tangent bundles and Poisson structures.
method Explicit construction of Lie groupoids and local models for symplectic integration.
result Necessary and sufficient topological condition for integration of elliptic tangent bundles.
Surveying integrability of Lie algebroids and structures.
problem Integrability of Lie algebroids and structures.
method Survey and recent results on integrability.
result Recent findings on local and global integrability.
Study integrability of generalized almost complex structures on S^6.
problem Integrability of generalized almost complex structures on the 6-dimensional sphere.
method Local coordinate criteria for integrability with respect to brackets and Courant integrability for strong structures.
result No nontrivial spherical combinations of the canonical structures are integrable with respect to the Levi-Civita connection.
Study introduces semi-integrable almost hyperhermitian structures.
problem Characterizing and constructing almost hyperhermitian structures.
method Divided structures into four classes and constructed on Lie algebras.
result Found semi-integrable structures on all compact Lie algebras of dimension 4n.
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a "Lagrangian" sub-2-groupoid of this symplectic 2-groupoid. As a corollary, we recover a result of Bursztyn-Crainic-Weinstein-Zhu that every integrabl…
Coupling Dirac structures are Dirac structures defined on the total space of a fibration, generalizing hamiltonian fibrations from symplectic geometry, where one replaces the symplectic structure on the fibers by a Poisson structure. We study the associated Poisson gauge theory, in order to describe the presymplectic g…
This paper describes integral affine structures on compact 3-manifolds.
problem Understanding integral affine structures on compact 3-manifolds.
method Analyzing complete integral affine structures on compact 3-manifolds up to finite-sheeted coverings.
result A complete list of integral affine structures on the three-dimensional torus and compact three-dimensional nilmanifolds was obtained.
Constructs integrable systems for Lie-Poisson structures at nilpotent elements.
problem Integrability of transverse Lie-Poisson structures at nilpotent elements.
method Using the argument shift method to construct families of functions in involution.
result Provides a uniform construction of completely integrable systems for an infinite family of nilpotent elements.
This paper explores integrability conditions for generalized metrics and structures on manifolds.
problem Investigating integrability conditions for generalized metrics and structures on manifolds.
method Considered two notions of integrability: Courant bracket and connection-induced bracket. Provided sufficient criteria for integrability.
result Sufficient criteria for integrability of generalized metrics and structures are formulated.
Defines new bi-flat structures from integrable systems and flat coordinates.
problem Creating new bi-flat structures from integrable systems.
method Combining Frölicher-Nijenhuis bicomplex with Lauricella bi-flat structures.
result Defines multi-parameter families of Lauricella bi-flat structures.
In 5D, integrability is linked to curvature constraints of subconformal structures.
problem Dispersionless integrability in 5D partial differential equations.
method Relating integrability to curvature constraints of subconformal structures.
result In 5D, integrability is characterized by the vanishing of a certain curvature of the subconformal structure.
New integrable deformations for topological hierarchies from Frobenius manifolds.
problem Integrable deformations of topological hierarchies from Frobenius manifolds.
method Construction of integrable deformations with polynomial tau-structures.
result Conjecture of universal object for Riemann--Hopf hierarchy.
Study integrability of specific geometric structures on odd Courant algebroids.
problem Characterize integrability of B_n-generalized structures on odd exact Courant algebroids.
method Characterize integrability in terms of existence of adapted generalized connections.
result Describe affine spaces of adapted generalized connections for integrable structures.
Investigates new F-structures and their Cauchy-Riemann properties.
problem Exploring new F-structures satisfying specific polynomial conditions. method Analyzes Cauchy-Riemann structure and integrability conditions.
result Identifies conditions for partial and complete integrability of F-structures. The paper proves stability of certain singularities in integrable systems.
problem Stability of singularities in integrable systems under perturbations.
method Analytic and smooth perturbations of completely integrable systems, connectedness condition.
result Non-degenerate singular fibers are structurally stable under small perturbations.
This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-stru…
New integrators preserve geometric structure in Hamiltonian systems.
problem Preserving geometric structure in Hamiltonian systems on Jacobi manifolds.
method Combining Poissonization and symplectic bi-realizations to construct structure-preserving integrators.
result Explicit construction and application of Jacobi Hamiltonian integrators.
Study integrability in Poisson and Dirac structures from quotients.
problem Integrability of quotient constructions in Poisson and Dirac geometry.
method Analysis of Poisson and Dirac structures from quotient constructions.
result Explicit constructions of Lie groupoids integrating specific geometric structures.
Study on quaternionic contact structures with integrable complementary distribution.
problem Characterize quaternionic contact structures with integrable complementary distributions.
method Analyze positive definite quaternionic contact (4n+3)-manifolds, focusing on the integrable complementary distribution and its relationship with Sasaki and 3-Sasaki structures. result Identify a new class of quaternionic contact structures with integrable complementary distributions that are not isomorphic to su(2). New curvature equations obstruct integrability of complex structures.
problem Understanding curvature obstructions to integrability of complex structures.
method Direct approach using Nijenhuis tensor derivatives and curvature scalars.
result Certain complex structures cannot coexist with non-flat constant curvature metrics.
This paper integrates Nijenhuis structures into Lie groupoids.
problem Understanding Nijenhuis structures and their global counterparts.
method Identifying and integrating Lie algebroids and Lie groupoids.
result Nijenhuis structures can be integrated to Lie groupoids and vice versa.
The paper shows inequality and rigidity for manifolds with integral Ricci curvature.
problem Analyzing structures of manifolds with integral Ricci curvature.
method Using segment inequality and similar methods as in \cite{CC1}, derive almost rigidity structure results.
result Sharp Hölder continuity result holds in the limit space of manifolds with integral Ricci curvature bound.
Researchers confirm integral formulas for G2-structures in detail.
problem Verifying integral formulas for G2-structures on Riemannian manifolds. method Detailed analysis and explicit expressions of intrinsic torsion using exterior algebra.
result Agreement of integral formulas with intrinsic torsion components.
Conditions for metallic structures to induce integrable distributions and geodesic invariance.
problem Conditions for metallic structures to induce integrable and geodesically invariant distributions.
method Analyzing tensor fields and adapted connections associated with metallic structures.
result Chen-type inequality for metallic distributions and conditions for metallic maps to preserve these distributions.
We discuss the integrability of rank 2 sub-Riemannian structures on low-dimensional manifolds, and then prove that some structures of that type in dimension 6, 7 and 8 have a lot of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing fields and the Hamiltonian, thus …
Characterizes integrability of generalized structures on Courant algebroids.
problem Integrability of generalized structures on Courant algebroids.
method Characterization via torsion-free generalized connections and Dirac generating operators.
result Criterion for integrability of generalized almost Hermitian structures and hyper-Hermitian structures.
Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.
problem Characterizing the geometry of solutions to 4D integrable equations.
method Defining rational normal scrolls and showing their involutivity.
result Involutive scroll structures are governed by a dispersionless integrable hierarchy.
Study complex structures with totally real sections, providing integrability equations.
problem Existence and integrability of complex structures with totally real sections.
method Explicit integrability equations derived from fiberwise Taylor expansions.
result Detailed fiberwise Taylor expansions and integrability equations in a geometric case.
A complex structure on a subset of S^6 cannot be extended to a global integrable structure.
problem Non-integrability of complex structures on subsets of S^6.
method Proving the non-integrability of extensions of a specific complex structure on a subset of S^6.
result It is impossible to deform a non-integrable structure to an integrable one on S^6 while fixing it on a subset.
Proves modular functors for SO(3) have integral Hodge structures.
problem Proving modular functors have integral Hodge structures.
method Based on homological models and geometric identification.
result Geometric construction of Hodge structures on SO(3) modular functors.
We classify linear Nambu structures (which are generalized Poisson structures in Hamiltonian dynamics and which give rise to integrable differential forms and singular foliations), then give a linearization for Nambu structures anf integrable differential forms near a nondegenerate singular point.
Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
Study of geometric structures on manifolds, focusing on integrability conditions.
problem Understanding the integrability of specific geometric structures.
method Analysis of algebraic types, intrinsic torsions, and distinguished connections.
result Presented first-order integrability conditions and geometric interpretations.
We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the…
Paper studies invariant distributions of bi-Hamiltonian structures.
problem Integrability of invariant distributions in bi-Hamiltonian structures.
method Description and investigation of invariant distributions.
result All invariant distributions of non-degenerate bi-Hamiltonian structures are described.
Integrability conditions for complex structures on product twistor spaces linked to Weyl geometry.
problem Integrability conditions for complex structures on product twistor spaces.
method Using Hitchin's generalized geometry and a metric connection with skew-symmetric torsion, the integrability conditions are derived in terms of Weyl geometry.
result Examples of structures satisfying the integrability conditions are supplied.
Characterizes almost abelian Lie algebras with integrable complex structure
problem Classifying almost abelian Lie algebras
method Using presentations consisting of a real number, an element in a vector space, and an endomorphism
result Classifies p-Kähler, p-pluriclosed, Kähler, balanced, pluriclosed, and Gauduchon metrics We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution of multi-dimensional second order partial differential equations (PDEs). We use Vessiot theory and solvable structures to find the largest integrable distributions contained in the Vessiot distribution assoc…
We classify all integrable complex structures on 6-dimensional Lie algebras of the form g×g.
This paper explains the fundamental relation between Jacobi structures and the classical Spencer operator coming from the theory of PDEs so as to provide a direct and geometric approach to the integrability of Jacobi structures. It uses recent results on the integrability of Spencer operators and multliplicative forms …
In this paper we interpret the integrability of the Dirac structures on some Hilbert C*-modules in terms of an automorphism group. This is the group of orthogonal transformations on the Hilbert C*-module of sections of a Hermitian vector bundle over an smooth manifold M. Some topological properties of the group of inte…
New integrators for mechanical systems on Lie groups simplify based on group properties.
problem Designing numerical integrators for mechanical systems on Lie groups.
method Leverage retraction maps and Lie group properties to design structure-preserving integrators.
result Simplified design of integrators for Euler-Poincare and Lie-Poisson equations.
We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a weakly holomorphic symplectic groupoid, which is a real symplectic groupoid with a compatible complex structure defined only on the associated stack, i.e., only up to Morita equivalence. We explain how such…
We expose some ideas from mathematical logics, i.e. the background of the theory of o-minimal structures, and demonstrate how they lead to the notion of a tame integral of motion and some extensions and clarifications of previous results on obstructions to integrability of geodesic flows.
We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence Kähler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant two-forms implies the integrability of the almost complex structure. This proves the con…
We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.
We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.