Frobenius integrability and Finsler metrizability for 2-dimensional sprays

Bucataru, Ioan; Creţu, Georgeta; Taha, Ebtsam H.

doi:10.1016/j.difgeo.2017.10.002

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…Definition 3.1. [5] Let (M, S) be a 2-dimensional smooth manifold M equipped with a nonflat spray S. Let H ∈ X(T M ) be a positive 2-homogeneous horizontal vector field such that β(H) = 0, where β is semi-basic 1-form defined in Definition 2.2. The regular 3-dimensional distribution defined by D = span{S, H, V } is called Berwald distribution.…”

Section: Berwald Frame and Berwald Connection On Finsler Surfacesmentioning

confidence: 99%

“…Lemma 3.2. [5] Let S be the geodesic spray of a Finsler function F and let D be its Berwald distribution. Then, we have…”

Section: Berwald Frame and Berwald Connection On Finsler Surfacesmentioning

confidence: 99%

“…Such a frame has been studied and utilized in [11, §9.9.1]. The Berwald frame (S, C, H, V ) has been defined directly for an arbitrary nonflat spray S and its properties applied to obtain information about the Finsler metrizability of the given spray S in [5]. The Berwald distribution D is a 3-dimensional distribution defined by D = span{S, H, V }, whose integrability provides a candidate for the Finsler function F .…”

Section: Introductionmentioning

confidence: 99%

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On Finsler surfaces with certain flag curvatures

Taha¹

2020

Preprint

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In the present paper, we investigate some geometric objects associated with the global Berwald distribution D := span{S, H, V := JH} of a 2-dimensional Finsler metrizable nonflat spray S. We find out necessary and sufficient conditions for a Finsler surface (M, F ) to be Landsbregian in terms of the Berwald connection 2-forms. Then, we study Finsler surfaces which satisfy some flag curvature K conditions, viz., V (K) = 0, V (K) = −I/F 2 and V (K) = −I K, using the global Berwald distribution. We obtain some classifications of such surfaces and show that under what hypothesis these surfaces turn to be Riemannian.

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Section: Berwald Frame and Berwald Connection On Finsler Surfacesmentioning

confidence: 99%

“…Lemma 3.2. [5] Let S be the geodesic spray of a Finsler function F and let D be its Berwald distribution. Then, we have…”

Section: Berwald Frame and Berwald Connection On Finsler Surfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Finsler surfaces with certain flag curvatures

Taha¹

2020

Preprint

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On Finsler surfaces with certain flag curvatures

Taha

2022

Int. J. Geom. Methods Mod. Phys.

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We find necessary and sufficient conditions for a Finsler surface [Formula: see text] to be Landsbregian in terms of the Berwald curvature [Formula: see text]-form. We study Finsler surfaces which satisfy some flag curvature [Formula: see text] conditions, viz., [Formula: see text] and [Formula: see text] where [Formula: see text] is the Cartan scalar. In order to do so, we investigate some geometric objects associated with the global Berwald distribution [Formula: see text] of a [Formula: see text]-dimensional Finsler metrizable nonflat spray [Formula: see text]. We find out under what conditions these surfaces turn out to be Riemannian. The existence of a first integral for the geodesic flow in each case has some remarkable consequences concerning rigidity results. We prove that a Finsler surface with [Formula: see text] and either [Formula: see text] or [Formula: see text] is Riemannian. Further, a Finsler surface with [Formula: see text] and [Formula: see text] is Riemannian.

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The geometry of geodesic invariant functions and applications to Landsberg surfaces

Elgendi,

Muzsnay

2024

MATH

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<p>In this paper, for a given spray $ S $ on an $ n $-dimensional manifold $ M $, we investigated the geometry of $ S $-invariant functions. For an $ S $-invariant function $ {\mathcal P} $, we associated a vertical subdistribution $ {{\mathcal V}}_{\mathcal P} $ and found the relation between the holonomy distribution and $ {{\mathcal V}}_{\mathcal P} $ by showing that the vertical part of the holonomy distribution is the intersection of all spaces $ {{\mathcal V}}_{ {\mathcal F}_S} $ associated with $ {\mathcal F}_S $ where $ {\mathcal F}_S $ is the set of all Finsler functions that have the geodesic spray $ S $. As an application, we studied the Landsberg Finsler surfaces. We proved that a Landsberg surface with $ S $-invariant flag curvature is Riemannian or has a vanishing flag curvature. We showed that for Landsberg surfaces with non-vanishing flag curvature, the flag curvature is $ S $-invariant if and only if it is constant; in this case, the surface is Riemannian. Finally, for a Berwald surface, we proved that the flag curvature is $ H $-invariant if and only if it is constant.</p>

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Frobenius integrability and Finsler metrizability for 2-dimensional sprays

Cited by 3 publications

References 15 publications

On Finsler surfaces with certain flag curvatures

On Finsler surfaces with certain flag curvatures

On Finsler surfaces with certain flag curvatures

The geometry of geodesic invariant functions and applications to Landsberg surfaces

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