Nijenhuis Geometry

Resumen:

The talk is devoted to Nijenhuis operators, this is fields of endomorphisms with vanishing Nijenhuis torsion. This is possibly the simplest geometric condition on fields of endomorphisms, for this reason it appeared time and again in geometry, analysis and mathematical physics. In particular it is relevant in the theory of projectively equivalent metrics and in the theory of integrable systems.

The talk is based on a series of papers aimed at redirecting the research agenda in this area to next level topics like:

Singular points: What does it mean for a point to be generic or singular in the context of Nijenhuis geometry? What singularities are non-degenerate, what are stable? How do Nijenhuis operators behave near non-degenerate or stable singular points?

Global properties: What restrictions on a Nijenhuis operator are imposed by compactness of the underlying manifold? Conversely, what are topological obstructions on a manifold carrying a Nijenhuis operator with specific properties, for example with no singular points? 

We demostrate that this research program is realistic by proving a series of new, not at all obvious results.

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