Non-existence of Funk functions for Finsler spaces of non-vanishing scalar flag curvature

Ioan Bucataru; Zoltán Muzsnay

doi:10.1016/j.crma.2016.04.001

Differential geometry

Non-existence of Funk functions for Finsler spaces of non-vanishing scalar flag curvature

Presented by: the Editorial Board

Ioan Bucataru ¹; Zoltán Muzsnay ²

¹ Faculty of Mathematics, Alexandru Ioan Cuza University, Iaşi, Romania
² Institute of Mathematics, University of Debrecen, Debrecen, Hungary

Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 619-622.

Abstract
Résumé

In his book Differential Geometry of Spray and Finsler spaces, page 177, Zhongmin Shen asks “whether or not there always exist non-trivial Funk functions on a spray space”. In this note, we will prove that the answer is negative for the geodesic spray of a Finslerian function of non-vanishing scalar flag curvature.

Dans son livre Differential Geometry of Spray and Finsler spaces, page 177, Zhongmin Shen demande s'il existe toujours une fonction de Funk non triviale sur un espace de spray. Dans la présente note, nous prouvons que la réponse est négative pour le spray géodésique d'une fonction finslérienne de courbure scalaire non nulle.

Received: 2015-06-03
Accepted: 2016-04-05
Published online: 2016-04-14

DOI: 10.1016/j.crma.2016.04.001

Author's affiliations:

Ioan Bucataru ¹; Zoltán Muzsnay ²

¹ Faculty of Mathematics, Alexandru Ioan Cuza University, Iaşi, Romania
² Institute of Mathematics, University of Debrecen, Debrecen, Hungary

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     author = {Ioan Bucataru and Zolt\'an Muzsnay},
     title = {Non-existence of {Funk} functions for {Finsler} spaces of non-vanishing scalar flag curvature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {619--622},
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Ioan Bucataru; Zoltán Muzsnay. Non-existence of Funk functions for Finsler spaces of non-vanishing scalar flag curvature. Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 619-622. doi : 10.1016/j.crma.2016.04.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.04.001/

References
Cited by

[1] I. Bucataru Funk functions and projective deformations of sprays and Finsler spaces of scalar flag curvature, J. Geom. Anal. (2016) | DOI

[2] I. Bucataru; Z. Muzsnay Projective and Finsler metrizability: parametrization rigidity of geodesics, Int. J. Math., Volume 23 (2012) no. 6 (15 pages)

[3] J. Grifone; Z. Muzsnay Variational Principles for Second-Order Differential Equations, World Scientific, 2000

[4] Z. Shen Differential Geometry of Spray and Finsler Spaces, Springer, 2001

[5] J. Szilasi A setting for spray and Finsler geometry (P.L. Antonelli, ed.), Handbook of Finsler Geometry, vol. 2, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003, pp. 1183-1426

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