An Account on Links Between Finsler and Lorentz Geometries for Riemannian Geometers

Javaloyes, Miguel Ángel; Pendás-Recondo, Enrique; Sánchez, Miguel

doi:10.1007/978-3-031-39916-9_10

Cited by 3 publications

(1 citation statement)

References 61 publications

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“…In the past decade these curves and transformations have been used successfully to model of a wide variety of natural shapes, from plant leaves and tree rings to starfish and avian eggs [5]. Further applications of Finsler geometry in the natural sciences involving superellipses or Gielis curves, are found in forest ecology [6,7], seismic ray paths in anisotropic media [8], and the spreading of wildfires [9]. This has inspired various researchers to study various geometrical transformations, including inversions [10], whereby the inversion occurs with Lamé-Gielis curves as inversion circles.…”

Section: Introductionmentioning

confidence: 99%

A New Approach for the Circular Inversion in l1- Normed Spaces

Ermiş,

Şen,

Gielis

2024

Preprint

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While there are well-known synthetic methods in the literature to find the image of a point under circular inversion in l2−normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometry of finite-dimensional Banach spaces. In this study, we have succeeded in giving a synthetic construction for the circular inversion in l1−normed spaces, which is one of the most fundamental examples of Minkowski geometry. Moreover, this synthetic construction has been given using the Euclidean circle, independently of the l1−norm.

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Section: Introductionmentioning

confidence: 99%

A New Approach for the Circular Inversion in l1- Normed Spaces

Ermiş,

Şen,

Gielis

2024

Preprint

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Globally hyperbolic spacetimes: slicings, boundaries and counterexamples

Sánchez

2022

Gen Relativ Gravit

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On the metrizability ofm-Kropina spaces with closed null one-form

Heefer

Pfeifer

Voorthuizen

et al. 2023

Journal of Mathematical Physics

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We investigate the local metrizability of Finsler spaces with m-Kropina metric F = α1+ m β− m, where β is a closed null one-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric α and one-form β have a very specific form in certain coordinates. In particular, when the signature of α is Lorentzian, α belongs to a certain subclass of the Kundt class and β generates the corresponding null congruence, and this generalizes in a natural way to arbitrary signature. We use this result to prove that the affine connection on such an m-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed from the affine connection is symmetric. In particular, we construct all counterexamples of this type to Szabo’s metrization theorem, which has only been proven for positive definite Finsler metrics that are regular on all of the slit tangent bundle.

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An Account on Links Between Finsler and Lorentz Geometries for Riemannian Geometers

Cited by 3 publications

References 61 publications

A New Approach for the Circular Inversion in l1- Normed Spaces

A New Approach for the Circular Inversion in l1- Normed Spaces

Globally hyperbolic spacetimes: slicings, boundaries and counterexamples

On the metrizability ofm-Kropina spaces with closed null one-form

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