Finsler metrics on surfaces admitting three projective vector fields

Lang, Julius

doi:10.1016/j.difgeo.2019.101590

Cited by 2 publications

(2 citation statements)

References 12 publications

(15 reference statements)

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“…As mentioned above, 2-dimensional projective structures are always variational. J. Lang in [36] constructed Lagrangians for 2-dimensional path and projective structures with the submaximal symmetry algebra (of dimension 3), see also [30,50].…”

Section: Resultsmentioning

confidence: 99%

Almost every path structure is not variational

Kruglikov

Matveev

2022

Gen Relativ Gravit

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Given a smooth family of unparameterized curves such that through every point in every direction there passes exactly one curve, does there exist a Lagrangian with extremals being precisely this family? It is known that in dimension 2 the answer is positive. In dimension 3, it follows from the work of Douglas that the answer is, in general, negative. We generalise this result to all higher dimensions and show that the answer is actually negative for almost every such a family of curves, also known as path structure or path geometry. On the other hand, we consider path geometries possessing infinitesimal symmetries and show that path and projective structures with submaximal symmetry dimensions are variational. Note that the projective structure with the submaximal symmetry algebra, the so-called Egorov structure, is not pseudo-Riemannian metrizable; we show that it is metrizable in the class of Kropina pseudo-metrics and explicitly construct the corresponding Kropina Lagrangian.

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Section: Resultsmentioning

confidence: 99%

Almost every path structure is not variational

Kruglikov

Matveev

2022

Gen Relativ Gravit

View full text Add to dashboard Cite

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“…As mentioned above, 2-dimensional projective structures are always variational. J. Lang in [34] constructed Lagrangians for 2-dimensional path and projective structures with the submaximal symmetry algebra (of dimension 3), see also [28,46].…”

mentioning

confidence: 99%

Almost every path structure is not variational

Kruglikov,

Matveev

2022

Preprint

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Given a smooth family of unparameterized curves such that through every point in every direction there passes exactly one curve, does there exist a Lagrangian with extremals being precisely this family? It is known that in dimension 2 the answer is positive. In dimension 3, it follows from the work of Douglas that the answer is, in general, negative. We generalise this result to all higher dimensions and show that the answer is actually negative for almost every such a family of curves, also known as path structure or path geometry.On the other hand, we consider path geometries possessing infinitesimal symmetries and show that path and projective structures with submaximal symmetry dimensions are variational. Note that the projective structure with the submaximal symmetry algebra, the so-called Egorov structure, is not pseudo-Riemannian metrizable; we show that it is metrizable in the class of Kropina pseudo-metrics and explicitly construct the corresponding Kropina Lagrangian.

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Finsler metrics on surfaces admitting three projective vector fields

Abstract: We show that in dimension 2 every Finsler metric with at least 3-dimensional Lie algebra of projective vector fields is locally projectively equivalent to a Randers metric. We give a short list of such Finsler metrics which is complete up to coordinate change and projective equivalence.

Cited by 2 publications

References 12 publications

Almost every path structure is not variational

Almost every path structure is not variational

Almost every path structure is not variational

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