Nijenhuis Geometry: singular points and applications

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Our long-term goal is to create a new branch of mathematics, Nijenhuis Geometry, with many applications in differential geometry and mathematical physics. Nijenhuis geometry studies Nijenhuis operators, i.e., fields of endomorphisms (or (1,1)-tensor fields) on a smooth manifold such the Nijenhuis torsion vanishes. The project consists of two parts, general theory of Nijenhuis operators and their applications. The main objectives of the first part are: (A) Local description: to what ‘normal’ form can one bring a Nijenhuis operator near every or almost every point by a local coordinate change? (B) Singular points: what does it mean for a point to be generic or singular in the context of Nijenhuis geometry? What singularities are typical? Non-degenerate? Stable? How do Nijenhuis operators behave near non-degenerate and stable singular points? (C) Global properties: what restrictions on a Nijenhuis operator are imposed by compactness of the manifold? Conversely, what are topological obstructions for a manifold carrying a Nijenhuis operator with specific properties (e.g. with no singular points, or with singular points of a prescribed type)? For the second “application’’ part, we selected the following three topics where Nijenhuis operators naturally appear and where our previously obtained results and the results obtained within this project are expected to help: a. Geodesically equivalent metrics. b. Infinite dimensional integrable systems of hydrodynamic type. c. Compatible infinite-dimensional Poisson brackets and related multicomponent integrable systems.