Nijenhuis forms help understand deformations of Lie algebroids and Poisson structures.
problem Understanding deformations of Lie algebroids and related structures.
method Introducing Nijenhuis forms on Lie-infinity algebras.
result Nijenhuis forms provide a new perspective on Poisson-Nijenhuis and quasi-Nijenhuis structures.
New expressions for Nijenhuis tensor squares found.
problem Understanding Nijenhuis tensor squares.
method Expressed dual forms in terms of exterior derivative derivatives.
result New vanishing results for Nijenhuis tensor squares.
Study on Nijenhuis tensor forms and vanishing properties.
problem Understanding the properties of Nijenhuis tensor.
method Provided strong and weak forms of the square of Nijenhuis tensor, and identified vanishing results.
result Identified new vanishing results for the square of Nijenhuis tensor.
The study classifies and normalizes 3D gl-regular Nijenhuis operators.
problem Classifying and normalizing 3D gl-regular Nijenhuis operators.
method Classification and normal form proof.
result Proved A. Bolsinov's conjecture.
Study new involutivity theorems for Poisson quasi-Nijenhuis manifolds.
problem Understanding involutivity in Poisson quasi-Nijenhuis geometry.
method Present new versions of deformation and involutivity theorems under specific factorization hypotheses.
result New versions of involutivity theorems for Poisson quasi-Nijenhuis manifolds.
This paper studies gl-regular Nijenhuis operators and their properties.
problem Characterizing and understanding gl-regular Nijenhuis operators.
method Analyzing the properties of gl-regular Nijenhuis operators and proving their existence in a coordinate system.
result Discoveries of normal forms for singular points and topological restrictions for gl-regular Nijenhuis operators on closed surfaces.
In the first chapter, we give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry, by considering differential stacks as being Lie groupoids up to Morita equivalence. We prove the…
Develops Schouten-Nijenhuis bracket on infinite-dimensional manifolds.
problem Defining the Schouten-Nijenhuis bracket on smooth infinite-dimensional manifolds.
method Two-step approach: first for summable multivector fields, then for sections of a specific sheaf.
result Formalizes Schouten-Nijenhuis bracket on infinite-dimensional manifolds.
Deforms Poisson quasi-Nijenhuis manifolds using closed 2-forms.
problem Deforming Poisson quasi-Nijenhuis manifolds.
method Using quasi-Lie bialgebroids and twisting.
result Interprets deformation in terms of quasi-Lie bialgebroids and Courant algebroids.
In commutative differential geometry the Frölicher-Nijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In \cit!{3} the Frölicher-Nijenhuis bracket was developped for universal differential forms of non-commutative algebras, and several applications were given. In this paper this bracke…
This paper studies Lie groupoids and their vector-valued forms.
problem Understanding vector-valued forms on Lie groupoids.
method Examining multiplicative vector-valued forms and their graded Lie subalgebra structure.
result Multiplicative vector-valued forms on Lie groupoids form a graded Lie subalgebra.
Study of singularities in two-dimensional Nijenhuis operators with non-zero trace differential.
problem Characterizing singularities of two-dimensional Nijenhuis operators.
method Analyzing the smoothness of functions related to the determinant of the operator.
result Complete description of singularities for certain function classes.
The paper studies Nijenhuis operators with a unity and their connection to F-manifolds.
problem Understanding Nijenhuis operators and their relationship to F-manifolds.
method Established a Splitting Theorem for Nijenhuis operators with a unity and proved their equivalence to F-manifolds.
result The class of regular F-manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity.
A Kaehler-Nijenhuis manifold is a Kaehler manifold M, with metric g, complex structure J and Kaehler form F, endowed with a Nijenhuis tensor field A that is compatible with the Poisson stucture defined by F in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if either AJ=JA or AJ=-JA, M is …
New theory allows simultaneous block-diagonalization of commuting operator fields.
problem Normal forms of operator fields.
method Generalized Nijenhuis torsions and generalized Haantjes algebra.
result Simultaneous block-diagonalization of commuting operator fields.
We investigate Nijenhuis deformations of L∞-algebras, a notion that unifies several Nijenhuis deformations, namely those of Lie algebras, Lie algebroids, Poisson structures and Courant structures. Additional examples, linked to Lie n-algebras and n-plectic manifolds, are included.
The study describes Nijenhuis operators with specific properties.
problem Characterizing Nijenhuis operators with functional independence and determinant constraints.
method Proving the general form and describing the specific case of Nijenhuis operators.
result Complete description of Nijenhuis operators with nondegenerate determinant.
We prove that on any symplectic manifold whose symplectic form represents a rational cohomology class there exists a sequence of compatible almost complex structures whose Nijenhuis energy (the L2-norm of the Nijenhuis tensor) tends to zero. The sequence is obtained by stretching the neck around a Donaldson hypersur…
The paper introduces new structures for left-symmetric algebroids.
problem Developing new mathematical structures for left-symmetric algebroids.
method Introducing Koszul-Vinberg-Nijenhuis structures and related concepts.
result Koszul-Vinberg-Nijenhuis structures provide a hierarchy of structures.
New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.
problem Developing integrable systems from tensor field properties.
method Using Frölicher-Nijenhuis brackets to generate bi-differential graded algebras and PDE systems.
result New integrable nonlinear matrix PDEs and systems are derived.
We define covariant Lie derivatives acting on vector-valued forms on Lie algebroids and study their properties. This allows us to obtain a concise formula for the Frölicher-Nijenhuis bracket on Lie algebroids.
The paper characterizes integrability of tensors on manifolds.
problem Analyzing integrability conditions for various tensor types on manifolds.
method Analytical and geometric characterizations of integrability for different tensor types, using Nijenhuis tensors.
result Integrability of tensors is equivalent to algebraic constancy coupled with vanishing of Nijenhuis-type tensors.
This paper explores the relationship between Leibniz algebras and Nijenhuis operators.
problem Understanding the relationship between Leibniz algebras and Nijenhuis operators.
method Investigation of Nijenhuis operators on Leibniz algebras and classification of Leibniz bialgebras.
result Leibniz algebras are closely related to Nijenhuis operators, and triangular symplectic Leibniz bialgebras possess Nijenhuis operators.
We define the Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that to any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid is associated such a structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background…
The paper explores conditions for the existence of orthogonal almost complex structures on manifolds.
problem Conditions for the existence of orthogonal almost complex structures on manifolds.
method Analyzes the Nijenhuis tensor and its squared norm to determine the existence of orthogonal almost complex structures.
result There exists no orthogonal almost complex structure on the standard sphere \(S^6\) with \(|N|^2 < \frac{64}{5}\) everywhere.
The paper defines a new differential on manifolds with special holonomy and calculates their cohomology.
problem Defining and calculating cohomology on manifolds with special holonomy.
method Using the Frölicher-Nijenhuis bracket on parallel forms on G2- and mSpin(7)-manifolds. result Cohomology groups of differential forms and partial description of the cohomology of differential forms relative to the tangent bundle are calculated.
This paper compiles formulas involving differential operators and interior products.
problem Scattered identities in differential geometry involving various operators.
method Compilation and extension of formulas using the Schouten-Nijenhuis bracket and interior product.
result New formulas involving the de Rham codifferential and interior product.
Local formulas integrate multiplicative forms on Lie groupoids.
problem Integrating multiplicative forms on local Lie groupoids.
method Explicit formulas using infinitesimal data.
result Concrete integrations of various geometric structures.
Solves problem of describing transformations for upper triangular Toeplitz operators.
problem Describing coordinate transformations preserving upper triangular Toeplitz form of operator fields.
method Implicit formulas involving matrix-valued functions for describing transformations and Nijenhuis operators.
result Formulas for coordinate transformations and Nijenhuis operators in upper triangular Toeplitz form.
We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the …
We revisit our earlier work on the AKSZ formulation of topological sigma model on generalized complex manifolds, or Hitchin model. We show that the target space geometry geometry implied by the BV master equations is Poisson--quasi--Nijenhuis geometry recently introduced and studied by Stiénon and Xu (in the untwisted …
New brackets generalize Haantjes moduli and ensure integrability of operators.
problem Characterizing and integrating operators using Haantjes moduli.
method Introducing a new infinite class of brackets and proving integrability conditions.
result Vanishing of higher-level Nijenhuis torsions ensures integrability and block-diagonal form.
The paper studies invariant generalized complex structures on flag manifolds.
problem Classifying invariant generalized complex structures on flag manifolds.
method Analyzing invariant 4-dimensional generalized almost complex structures restricted to each root space, and studying the Nijenhuis operator for a triple of roots. result Classification of integrable and Ω-integrable generalized complex structures. 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.
Study torsion and curvature in ACYT and AHKT 8-manifolds.
problem Characterize torsion and curvature in ACYT and AHKT 8-manifolds.
method Analyzes properties of Nijenhuis tensors, Ricci tensors, and torsion forms on ACYT and AHKT 8-manifolds.
result Closed torsion condition and Ricci flatness for ACYT 8-manifolds.
Nijenhuis tensors N on Courant algebroids compatible with the pairing are studied. This compatibility condition turns out to be of the form N+N∗=aI for irreducible Courant algebroids, in particular for the extended tangent bundles TM⊕T∗M. It is proved that compatible Nijenhuis tensors on irreducible Coura…
The paper studies symmetries and conservation laws of non-diagonalisable hydrodynamic systems.
problem Integrating non-diagonalisable hydrodynamic systems of partial differential equations.
method Analysis of gl-regular Nijenhuis operators, splitting Theorem for symmetries and conservation laws, relationship between symmetries and conservation laws.
result The system of partial differential equations is integrable in quadratures.
The paper studies deformations of Nijenhuis structures in Lie algebras and algebroids.
problem Deformations of Nijenhuis structures in Lie algebras and algebroids.
method Operadic study, introduction of homotopy Nijenhuis Lie algebras, construction of L∞-algebras for deformations. result The Poincaré Lemma holds for certain Nijenhuis operators, confirming a conjecture.
Normal forms of almost complex structures in a neighborhood of pseudoholomorphic curve are considered. We define normal bundles of such curves and study the properties of linear bundle almost complex structures. We describe 1-jet of the almost complex structure along a curve in terms of its Nijenhuis tensor. For pseudo…
We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing the Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one…
We show how to reduce, under certain regularities conditions, a Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson-Nijenhuis manifolds. The choice of the more general framework of Lie a…
Introduces Lie-Nijenhuis bialgebroids for Poisson-Nijenhuis groupoids.
problem Describing Poisson-Nijenhuis groupoids infinitesimally.
method Develops a theory of generalized derivations and their duality.
result Lie-Nijenhuis bialgebroids provide a complete infinitesimal description of Poisson-Nijenhuis groupoids.
Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids.
New type of manifolds derived from Poisson structures.
problem Generalizing Poisson Nijenhuis manifolds.
method Introducing pseudo-Poisson Nijenhuis manifolds and showing their properties.
result Found new materials to construct Courant algebroids.
Establishes Morita equivalence for Nijenhuis structures and proves invariance of modular class.
problem Morita equivalence for Nijenhuis structures and Poisson-Nijenhuis manifolds.
method Global-to-infinitesimal correspondence using Lie functor and enhanced known equivalences.
result Modular class of Poisson-Nijenhuis manifolds is invariant under Morita equivalence.
The paper connects web theory to heavenly PDEs and Einstein metrics.
problem Understanding the correspondence between self-dual metrics and vector fields.
method Using Nijenhuis operators and web theory to construct new heavenly PDEs.
result New integrable heavenly PDEs derived from Nijenhuis operators.
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.
In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex supermanifolds, i.e., real supermanifolds with a non-integrable distribution, the c…