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.
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48 results for “Nijenhuis tensors”
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.
Small Nijenhuis tensor found on compact manifolds.
problem Finding compact manifolds with small Nijenhuis tensor.
method Provided explicit examples of manifolds with small Nijenhuis tensor.
result Examples of manifolds with small Nijenhuis tensor in various dimensions.
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…
Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
problem Understanding the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
method Reduction of computations to solving PDEs, explicit solutions on specific manifolds, analysis of curve of almost complex structures, classification of Lie algebras.
result Classification of Lie algebras admitting almost complex structures with specific Nijenhuis tensor ranks.
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.
An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. There are studied relations between the six associated Nijenhuis tensors as well as their vanishing. It is given a …
Introduces compatibility between Dirac structures and Nijenhuis tensors.
problem No specific problem stated; focuses on extending Poisson-Nijenhuis structures.
method Introduces compatibility between Dirac structures and (1,1)-tensor fields.
result Properties of Dirac-Nijenhuis structures studied, including connections and integrations.
We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.
Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, Poisson-Nijenhuis structures, and triangular …
Kaehler-Nijenhuis Manifoldsmath.DG
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 …
We extend to the context of Courant algebroids several hierarchies that can be constructed on Poisson-Nijenhuis manifolds. More precisely, we introduce several notions (Poisson-Nijenhuis, deformation-Nijenhuis and Nijenhuis pairs) that extend to Courant algebroids the notion of a Poisson-Nijenhuis manifold, by using th…
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.
Hydrodynamic hierarchy deformed using conservation laws.
problem Deforming a hydrodynamic hierarchy with non-vanishing Nijenhuis torsion.
method Using a chain of conservation laws to deform the hierarchy.
result The resulting hierarchy has non-vanishing Nijenhuis torsion but vanishing Haantjes tensor.
Maximally non-integrable almost complex structures: an -principle and cohomological propertiesmath.DG
Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.
problem Maximally non-integrable almost complex structures and their cohomological properties.
method h-principle and topological invariants characterization.
result Existence of almost complex structures with maximal Nijenhuis tensor rank on parallelizable and certain manifolds.
We generalize Poisson-Nijenhuis structures. We prove that on a manifold endowed with a Nijenhuis tensor and a Jacobi structure which are compatible, there is a hierarchy of pairwise compatible Jacobi structures. Furthermore, we study the homogeneous Poisson-Nijenhuis structures and their relations with Jacobi structure…
Poisson-NIjenhuis structures for an arbitrary Lie agebroid are defined and studied by means of tangent lifts of tensor fields.
Nijenhuis tensors on Courant algebroids compatible with the pairing are studied. This compatibility condition turns out to be of the form for irreducible Courant algebroids, in particular for the extended tangent bundles . It is proved that compatible Nijenhuis tensors on irreducible Coura…
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 -norm of the Nijenhuis tensor) tends to zero. The sequence is obtained by stretching the neck around a Donaldson hypersur…
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…
Study shows almost complex structures with certain tensor properties are prevalent.
problem Characterizing almost complex structures with specific tensor properties.
method Analyzes the space of almost complex structures on compact manifolds.
result The space of almost complex structures with rank at least k Nijenhuis tensor is either empty or dense in each component.
Research explores Lie algebras in Riemannian manifolds.
problem Understanding Lie algebras in Riemannian manifolds.
method Analyzes the Lie algebra of infinitesimal isometries.
result Identifies two commutative ideals in the Lie algebra.
Complete integrability from Poisson-Nijenhuis structures on compact hermitian symmetric spacesmath.SG
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 …
The study characterizes and proves properties of 3D Poisson quasi-Nijenhuis manifolds.
problem Characterizing and understanding 3D Poisson quasi-Nijenhuis manifolds.
method Characterization through deformation and application of Haantjes structures.
result Every 3D Poisson quasi-Nijenhuis manifold is a Haantjes manifold.
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.
Courant-Nijenhuis algebroidsmath.DG
Introduces new structures in geometry and algebra.
problem Understanding compatibility between geometric structures.
method Introduces Courant 1-derivations and Courant-Nijenhuis algebroids.
result Holomorphic Courant algebroids are a type of Courant-Nijenhuis algebroids.
Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we …
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 super or Z_2-graded Schouten-Nijenhuis bracket is introduced. Using it, new generalized super-Poisson structures are found which are given in terms of certain graded-skew-symmetric contravariant tensors Λof even order. The corresponding super `Jacobi identities' are expressed by stating that these tensors have zero…
The fundamental tool in the classification of orthogonal coordinate systems in which the Hamilton-Jacobi and other prominent equations can be solved by a separation of variables are second order Killing tensors which satisfy the Nijenhuis integrability conditions. The latter are a system of three non-linear partial dif…
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.
Let be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (so-called Nijenhuis tensor) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold is called nearly Kaehler if it adm…
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…
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.
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.
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.
Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis structure from a given type (1,1) tensor field J on Q. It is argued that the complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a cru…
We study natural lifting operations from a bundle E over R to the dual bundle of its first-jet bundle. The main purpose is to define a complete lift of a type (1,1) tensor field on E and to understand all features of its construction. Various other lifting operations of tensorial objects on E are needed for that purpos…
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.
Generalizations in many directions of the contraction procedure for Lie algebras introduced by E.J.Saletan are proposed. Products of arbitrary nature, not necessarily Lie brackets, are considered on sections of finite-dimensional vector bundles. Saletan contractions of such infinite-dimensional algebras are obtained vi…
Proposed theorems for lifts of the extended almost complex structures on the complex manifoldmath.DG
The paper studies lifts of complex structures on a manifold.
problem Understanding higher-order lifts of extended almost complex structures.
method Proved theorems on Nijenhuis tensor and introduced a new tensor field.
result Basic results on almost analytic complex vectors are investigated.
We study almost Hermitian structures admitting a Hermitian connexion with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connexio…
Banach fibrations and Nijenhuis operators studied for vanishing torsion.
problem Understanding Nijenhuis operators on Banach fibrations and their properties.
method Analyzing Nijenhuis operators on Banach fibrations and their projectability.
result Vanishing of Nijenhuis torsion on implies vertical values for .
In this paper we present some approaches to classification of almost complex structures and to construction of local or formal pseudoholomorphic mapping from one almost complex manifold to another. The corresponding criteria are given in terms of Nijenhuis tensors and their generalizations. We deal with the prolongatio…
Nijenhuis Geometrymath.DG
This work is the first, and main, of the series of papers in progress dedicated to Nienhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis tensor. It serves as an introduction to Nijenhuis Geometry that should be understood in much wider context than before: from local description at generic points t…
New integrable systems constructed for non-diagonal Killing tensors.
problem Constructing integrable Hamiltonian systems with quadratic momenta.
method Using Nijenhuis geometry and gl-regular Nijenhuis operators.
result Reproduces classical Stäckel construction and finds new systems for n≥3.
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.
Beyond recursion operatorsmath.HO
We briefly recall the history of the Nijenhuis torsion of (1,1)-tensors on manifolds and of the lesser-known Haantjes torsion. We then show how the Haantjes manifolds of Magri and the symplectic-Haantjes structures of Tempesta and Tondo generalize the classical approach to integrable systems in the bi-hamiltonian and s…