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

Bucataru, Ioan1 ; Muzsnay, Zoltán2
1 Faculty of Mathematics, Alexandru Ioan Cuza University, Iaşi, Romania
2 Institute of Mathematics, University of Debrecen, Debrecen, Hungary
Comptes Rendus. Mathématique, Tome 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
Bucataru, Ioan 1 ; Muzsnay, Zoltán 2

1 Faculty of Mathematics, Alexandru Ioan Cuza University, Iaşi, Romania
2 Institute of Mathematics, University of Debrecen, Debrecen, Hungary 
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Bucataru, Ioan; Muzsnay, Zoltán. Non-existence of Funk functions for Finsler spaces of non-vanishing scalar flag curvature. Comptes Rendus. Mathématique, Tome 354 (2016) no. 6, pp. 619-622. doi : 10.1016/j.crma.2016.04.001. http://www.numdam.org/articles/10.1016/j.crma.2016.04.001/

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

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

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

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

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

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