×

zbMATH — the first resource for mathematics

The multiplier approach to the projective Finsler metrizability problem. (English) Zbl 1257.53105
Summary: This paper is concerned with the problem of determining whether a projective-equivalence class of sprays is the geodesic class of a Finsler function. We address both the local and the global aspects of this problem. We present our results entirely in terms of a multiplier, that is, a type-(0,2) tensor field along the tangent bundle projection. In the course of the analysis we consider several related issues of interest including the positivity and strong convexity of positively-homogeneous functions, the relation to the so-called Rapcsák conditions, some peculiarities of the two-dimensional case, and geodesic convexity for sprays.

MSC:
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Álvarez Paiva, J.C., Symplectic geometry and hilbertʼs fourth problem, J. diff. geom., 69, 353-378, (2005) · Zbl 1088.53047
[2] Álvarez Paiva, J.C.; Berck, G., Finsler surfaces with prescribed geodesics · Zbl 1099.53018
[3] Bácsó, S.; Szilasi, Z., On the projective theory of sprays, Acta math. acad. paed. nyíregyháziensis, 26, 171-207, (2010) · Zbl 1240.53047
[4] Bao, D.; Chern, S.-S.; Shen, Z., An introduction to Riemann-Finsler geometry, (2000), Springer · Zbl 0954.53001
[5] Bryant, R.L., Projectively flat Finsler 2-spheres of constant curvature, Selecta math. (N. S.), 3, 161-203, (1997) · Zbl 0897.53052
[6] Bucataru, I.; Muzsnay, Z., Projective metrizability and formal integrability, Sigma, 7, 114, (2011) · Zbl 1244.49072
[7] Crampin, M., Some remarks on the Finslerian version of hilbertʼs fourth problem, Houston J. math., 37, 369-391, (2011) · Zbl 1228.53085
[8] Crampin, M.; Saunders, D.J., Path geometries and almost Grassmann structures, Adv. stud. pure math., 48, 225-261, (2007) · Zbl 1168.53010
[9] Douglas, J., Solution of the inverse problem of the calculus of variations, Trans. amer. math. soc., 50, 71-128, (1941) · JFM 67.1038.01
[10] Henneaux, M., On the inverse problem of the calculus of variations, J. phys. A: math. gen., 15, L93-96, (1982) · Zbl 0475.70024
[11] Krupka, D.; Sattarov, A.E., The inverse problem of the calculus of variations for Finsler structures, Math. slovaca, 35, 217-222, (1985) · Zbl 0585.53019
[12] Krupková, O.; Prince, G.E., Second order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations, (), 837-904 · Zbl 1236.58027
[13] Lovas, R.L., A note on Finsler-Minkowski norms, Houston J. math., 33, 701-707, (2007) · Zbl 1152.53014
[14] Sarlet, W., The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics, J. phys. A: math. gen., 15, 1503-1517, (1982) · Zbl 0537.70018
[15] Sarlet, W., Linear connections along the tangent bundle projection, (), 315-340 · Zbl 1208.53018
[16] Shen, Z., Differential geometry of spray and Finsler spaces, (2001), Kluwer · Zbl 1009.53004
[17] Szilasi, J., Calculus along the tangent bundle projection and projective metrizability, (), 527-546
[18] Szilasi, J.; Vattamány, Sz., On the Finsler metrizabilities of spray manifolds, Period. math. hungarica, 44, 81-100, (2002) · Zbl 0997.53056
[19] Tabachnikov, S., Remarks on magnetic flows and magnetic billiards, Finsler metrics, and a magnetic analogue of hilbertʼs fourth problem, (), 23-252
[20] Warner, F.W., Foundations of differentiable manifolds and Lie groups, (1971), Scott, Foresmann · Zbl 0241.58001
[21] Whitehead, J.H.C., Convex regions in the geometry of paths, Quart. J. math., 3, 33-42, (1932) · Zbl 0004.13102
[22] Whitehead, J.H.C., Convex regions in the geometry of paths - addendum, Quart. J. math., 4, 226-227, (1933) · Zbl 0007.36801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.