Errata for the paper “classes of Finsler spaces with (α, β)-metrics”

For a standard Finsler metric F on a manifold M, the domain is the whole tangent bundle T M and the fundamental tensor g is positive-definite. However, in many cases (for example, for the well-known Kropina and Matsumoto metrics), these two conditions hold in a relaxed form only, namely one has either a pseudo-Finsler metric (with arbitrary g) or a conic Finsler metric (with domain a "conic" open domain of T M).Our aim is twofold. First, we want to give an account of quite a few subtleties that appear under such generalizations, say, for conic pseudo-Finsler metrics (including, as a preliminary step, the case of Minkowski conic pseudo-norms on an affine space). Second, we aim to provide some criteria that determine when a pseudo-Finsler metric F obtained as a general homogeneous combination of Finsler metrics and one-forms is again a Finsler metric -or, more precisely, that the conic domain on which g remains positive-definite. Such a combination generalizes the known (↵, )-metrics in different directions. Remarkably, classical examples of Finsler metrics are reobtained and extended, with explicit computations of their fundamental tensors.

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“…We remark that in [41,42] similar computations to those of the following lemma have been carried out for (↵, )-metrics.…”

Section: Appendixmentioning

confidence: 99%

On the definition and examples of Finsler metrics

Javaloyes¹,

Sánchez²

2014

Annali Scuola Normale Superiore - Classe Di Scienze

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“…All the above mentioned applications require Finsler metrics of Lorentzian signature. While there exists a rich literature on positive definite Finsler metrics (and in particular, on (α, β)-ones, [20][21][22][23][24][25]), Lorentzian Finsler geometry is by far less understood and investigated. In this paper, we study for the first time in full generality two questions about Lorentzian (α, β)-Finsler structures, which have been just partially tackled in the literature (mostly only for very particular cases):…”

Section: Introductionmentioning

confidence: 99%

The Finsler spacetime condition for (α,β)-metrics and their isometries

Voicu¹,

Friedl-Szász²,

Popovici-Popescu³

et al. 2023

Preprint

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For the general class of pseudo-Finsler spaces with (α, β)-metrics, we establish necessary and sufficient conditions such that these admit a Finsler spacetime structure. This means that the fundamental tensor has Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future pointing timelike vectors is ensured. The identified (α, β)-Finsler spacetimes are candidates for applications in gravitational physics. Moreover, we completely determine the relation between the isometries of an (α, β)-metric and the isometries of the underlying pseudo-Riemannian metric a; in particular, we list all (α, β)-metrics which admit isometries that are not isometries of a.

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“…Important textbooks which contain sections about Randers manifolds are [1,2]. Among the many papers on the subject we cite only a few [8,12,14,15,16] which have a direct bearing on the subject matter of this paper. What these works have in common, and in fact what is common to the bulk of the literature on Finsler geometry, is that the analysis is almost entirely based on computations in local coordinates.…”

Section: Introductionmentioning

confidence: 99%

On the geometry of randers manifolds

Mestdag

Tóth

2002

Reports on Mathematical Physics

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Abstract. Randers manifolds are studied in the framework of the pullback bundle formalism, with the aid of intrinsic methods only. After checking a sufficient condition for a Randers manifold to be a Finsler manifold, we provide a systematic description of the Riemann-Finsler metric, the canonical spray, the Barthel endomorphism, the Berwald connection, the Cartan tensors and the Cartan vector field in this new setting. Finally, as an application of the new tools and geometric ideas developed here, we present an intrinsic proof of the celebrated theorem about a criterion for a Randers manifold to become a Berwald manifold.

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Errata for the paper “classes of Finsler spaces with (α, β)-metrics”

Cited by 3 publications

References 0 publications

On the definition and examples of Finsler metrics

On the definition and examples of Finsler metrics

The Finsler spacetime condition for (α,β)-metrics and their isometries

On the geometry of randers manifolds

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