Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics

Bolsinov, Alexey V.; Matveev, Vladimir S.

doi:10.1090/s0002-9947-2011-05187-1

Cited by 30 publications

(84 citation statements)

References 38 publications

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“…in the case of complex conjugate eigenvalues λ Segre type [4] Segre type [3,1] Segre type [2,2] Segre type [2,1,1] Segre type indicates the number and sizes of Jordan blocks with the same eigenvalue λ.…”

Section: Classification Resultsmentioning

confidence: 99%

“…Lemma 1 (Splitting Lemma, [2]). Let L be an affinor with zero Nijenhuis torsion on a manifold M, dimM = n. Suppose there exists a (non-holonomic) frame in which L takes block diagonal form,…”

Section: The Splitting Lemmamentioning

confidence: 99%

“…Since the case [2,1] gives no non-constant examples, we will concentrate on Segre type [3]. Then there exists a coordinate system where g andg 0 take the form…”

Section: Proofmentioning

confidence: 99%

“…Hamiltonian operators of the form (1.1) have subsequently been generalised in a whole variety of different ways (degenerate, non-homogeneous, higher order, multi-dimensional and non-local, see [19] for a review), however, until now very few classification results are available due to the complexity of the problem. In this paper we address the classification of 2D Hamiltonian operators of DubrovinNovikov type, 2) which are generated by a pair of flat metrics g,g, see [5,16,17]. The operator P will be called non-degenerate if the tensor g + λg is non-degenerate for generic values of the parameter λ (without any loss of generality we will assume both g andg to It is known that the vanishing of the obstruction tensor is necessary and sufficient for the existence of coordinates where the operator (1.2) takes constant coefficient form [5] (2D Darboux theorem).…”

Section: Introductionmentioning

confidence: 99%

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Hamiltonian Operators of Dubrovin-Novikov Type in 2D

2014

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First order Hamiltonian operators of differential-geometric type were introduced by Dubrovin and Novikov in 1983, and thoroughly investigated by Mokhov. In 2D, they are generated by a pair of compatible flat metrics g andg which satisfy a set of additional constraints coming from the skew-symmetry condition and the Jacobi identity. We demonstrate that these constraints are equivalent to the requirement thatg is a linear Killing tensor of g with zero Nijenhuis torsion. This allowed us to obtain a complete classification of ncomponent operators with n ≤ 4 (for n = 1, 2 this was done before). For 2D operators the Darboux theorem does not hold: the operator may not be reducible to constant coefficient form. All interesting (non-constant) examples correspond to the case when the flat pencil g,g is not semisimple, that is, the affinorgg −1 has non-trivial Jordan block structure. In the case of a direct sum of Jordan blocks with distinct eigenvalues we obtain a complete classification of Hamiltonian operators for any number of components n, revealing a remarkable correspondence with the class of trivial Frobenius manifolds modelled on H * (CP n−1 ).MSC: 37K05, 37K10, 37K25, 53D45.

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Section: Classification Resultsmentioning

confidence: 99%

Section: The Splitting Lemmamentioning

confidence: 99%

“…Since the case [2,1] gives no non-constant examples, we will concentrate on Segre type [3]. Then there exists a coordinate system where g andg 0 take the form…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hamiltonian Operators of Dubrovin-Novikov Type in 2D

2014

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“…The pair (const ·g, C 1 a+ C 2 g + C 3 λ⊗ λ) determines the foliations corresponding to the coordinate plaques (x 1 , x 2 ), x (2) ,. .…”

Section: 3mentioning

confidence: 99%

Degree of mobility for metrics of Lorentzian signature and parallel (0,2)-tensor fields on cone manifolds

Fedorova

Matveev

2013

Proceedings of the London Mathematical Society

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Abstract. Degree of mobility of a (pseudo-Riemannian) metric is the dimension of the space of metrics geodesically equivalent to it. We describe all possible values of the degree of mobility on a simply connected n-dimensional manifold of lorentz signature. As an application we calculate all possible differences between the dimension of the projective and the isometry groups. One of the main new technical results in the proof is the description of all parallel symmetric (0, 2)-tensor fields on cone manifolds of signature (n − 1, 2).

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