Nijenhuis Geometry

Abstract

This work is the first, and main, of a 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.

Applications of Nijenhuis geometry: Nondegenerate singular points of Poisson-Nijenhuis structures

“…This paper is a follow-up of our previous work [2] that can be considered as an introduction to Nijenhuis Geometry. Nijenhuis geometry studies Nijenhuis operators, i.e., fields of endomorphisms L = (L i j ) of the tangent space with vanishing Nijenhuis torsion: that is, for any vector fields ξ, η…”

“…The paper [2] suggests a programme and general strategy for studying such operators (and demonstrates that the programme is realistic by proving a series of non-trivial results). Important part of the strategy is the study of singular points of Nijenhuis operators.…”

“…The goal of this paper is to illustrate how methods, ideas and results of [2] work in describing singular points of more complicated geometric structures involving Nijenhuis operators as one of their ingredients (cf. [2, Sections 6.2 and 6.3]) and also to give a proof for Theorem 6.3 announced in [2].…”

“…Study of singularities of Nijenhuis and other related structures is an important part of the programme suggested in [2] (see also [11]). Actually, in most branches of geometry understanding of singular points of relevant geometric structures plays a central role.…”

“…Among singular points, we distinguish (see [2,Definition 2.10]) an important subclass of the so-called differentially non-degenerate singular points p ∈ M which are characterised by the property that the differentials of the functions tr L k , k = 1, . .…”