Nijenhuis Geometry

Abstract

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 to singularities and global analysis. The goal of the present paper is to introduce terminology, develop new important techniques (e.g., analytic functions of Nijenhuis operators, splitting theorem and linearisation), summarise and generalise basic facts (some of which are already known but we give new self-contained proofs), and more importantly, to demonstrate that the research programme proposed in the paper is realistic by proving a series of new, not at all obvious, results.

References in corpus (3)

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
• Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators
• Integrable matrix equations related to pairs of compatible associative algebras

Cited by in corpus (16)

• Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type
• Haantjes Algebras and Diagonalization
• Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures
• Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity
• Orthogonal separation of variables for spaces of constant curvature
• When a $(1,1)$-tensor generates separation of variables of a certain metric
• Weak metric structures on generalized Riemannian manifolds
• Singularities of two-dimensional Nijenhuis operators
• Quasi-Lie bialgebroids, Dirac structures and deformations of Poisson quasi-Nijenhuis manifolds
• Almost Differentially Nondegenerate Nijenhuis Operators
• Integrating Nijenhuis Structures
• Elementary Differential Singularities of Three-Dimensional Nijenhuis Operators
• Nijenhuis geometry of parallel tensors
• Nijenhuis operators on homogeneous spaces related to $C^{\ast }$-algebras
• Parallel differential forms of codegree two, and three-forms in dimension six
• Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions