Comptes Rendus
no. 6
Differential geometry
Non-existence of Funk functions for Finsler spaces of non-vanishing scalar flag curvature
[Non-existence de fonctions de Funk pour les espaces de Finsler de courbure scalaire non nulle]

Présenté par : the Editorial Board

Ioan Bucataru1 ; Zoltán Muzsnay2
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.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.04.001
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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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/

[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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