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

Bolsinov, Alexey V.; Konyaev, Andrey Yu.; Matveev, Vladimir S.

doi:10.48550/arxiv.2001.04851

Cited by 4 publications

(12 citation statements)

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“…This is again a question on gl-regular Nijenhuis operators. It is amazing that the answer turns out to be very similar to that given by Arnold: there is a very simple generic Nijenhuis perturbation of a Jordan block (see formula (6) and Proposition 5.1), which is unique and coincides exactly with the one given in [2]. All the others can be derived from this canonical one by solving a system of integrable PDEs.…”

Section: Outlook and Motivationmentioning

confidence: 53%

“…A research programme and general strategy for studying such operators were suggested in [5]. This paper is devoted to the next item of our agenda (after [5], [6], [15]) and is focused on Nijenhuis operators satisfying gl-regularity condition.…”

Section: Basic Definitions and Main Resultsmentioning

confidence: 99%

“…This paper, as well as its predecessors [5,6,15], aim to repair this situation. An important ingredient of our strategy described in [5] is to develop tools to study and describe Nijenhuis operators near those points where the Segre characteristic changes (singular points in the terminology of [5]) and also on closed manifolds.…”

Section: Outlook and Motivationmentioning

confidence: 99%

“…Proof. The existence of solutions of ( 16) for all initial conditions in a more general case is discussed in [6] (and, in fact, can be derived from the Cartan-Kähler theorem [12]). The necessary and sufficient condition is…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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Nijnehuis Geometry III: gl-regular Nijenhuis operators

Bolsinov,

Konyaev,

Matveev

2020

Preprint

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We study Nijenhuis operators, that is, (1, 1)-tensors with vanishing Nijenhuis torsion under the additional assumption that they are gl-regular, i.e., every eigenvalue has geometric multiplicity one. We prove the existence of a coordinate system in which the operator takes first or second companion form, and give a local describtion of such operators. We apply this local description to study singular points. In particular, we obtain their normal forms in dimension two and discover topological restrictions for the existence of gl-regular Nijenhuis operators on closed surfaces.This paper is an important step in the research programme suggested in [5,8].

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Section: Outlook and Motivationmentioning

confidence: 53%

Section: Basic Definitions and Main Resultsmentioning

confidence: 99%

Section: Outlook and Motivationmentioning

confidence: 99%

Section: Proof Of Theorem 11mentioning

confidence: 99%

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Nijnehuis Geometry III: gl-regular Nijenhuis operators

Bolsinov,

Konyaev,

Matveev

2020

Preprint

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“…This paper continues the Nijenhuis Geometry programme started in [1] and further developed in [2,3,20]. This programme was initially motivated by the fact that Niejnhuis operators (i.e.…”

Section: Introductionmentioning

confidence: 88%

Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

Bolsinov,

Konyaev,

Matveev

2020

Preprint

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We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n + 1)(n + 2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ≤ n + 2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n + 2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.

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