Geodesics of two-dimensional Finsler spaces. (English) Zbl 0812.53022

Summary: A concise description of 2-dimensional Finsler spaces is presented from the viewpoint of their geodesic curves (i.e., extremals of a variational problem). Berwald’s classification and geodesics of 1-form metric spaces are studied. Darboux’s solution to the problem of determination of the variational functional (homogeneous Lagrangian) from given geodesic equations is presented.


53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C22 Geodesics in global differential geometry
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[1] Antonelli, P.L., Finsler Volterra-Hamilton systems in ecology, Tensor, N.S., 50, 22-31, (1991) · Zbl 0758.53012
[2] M. Matsumoto, Geometrical aspect of P.L. Antonelli’s Finsler metric in his paper “Finsler Volterra-Hamilton systems in ecology”, Tensor, N.S. (to appear). · Zbl 0830.53021
[3] Dryuma, V., Finsler’s metric of the nonlinear dynamical system phase space and its applications, Rep. on IX internl, confer. “topology and its applications”, (1992), Kiev, Ukraine
[4] Matsumoto, M., The main scalar of two-dimensional Finsler spaces with special metrics, J. math. Kyoto univ., 32, 889-898, (1992) · Zbl 0784.53016
[5] R.S. Ingarden and L. Tamássy, On parabolic geometry and irreversible macroscopic time, Rep. on Math. Phys. (to appear).
[6] Antonelli, P.L.; Ingarden, R.S.; Matsumoto, M., The theory of sprays and Finsler spaces with applications in physics and biology, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0821.53001
[7] Asanov, G.S., Finsler geometry, relativity and gauge theories, (1985), D. Reidel Dordrecht, Holland · Zbl 0576.53001
[8] Asanov, G.S., Two-dimensional Finsler spaces, (1990), University of Athens, Sem. P. Zervos · Zbl 0444.53025
[9] Bejancu, A., Finsler geometry and applications, (1990), Ellis Horwood London
[10] Matsumoto, M., Foundations of Finsler geometry and special Finsler spaces, (1986), Kaiseisha Press Saikawa, Ōtsu, Japan · Zbl 0594.53001
[11] Matsumoto, M., The inverse problem of variation calculus in two-dimensional Finsler space, J. math. Kyoto univ., 29, 489-496, (1989) · Zbl 0696.53019
[12] Matsumoto, M., On Wagner’s generalized Berwald spaces of dimension two, Tensor, N.S., 36, 303-311, (1982) · Zbl 0508.53028
[13] Matsumoto, M.; Shimada, H., On Finsler spaces with 1-form metric, Tensor, N.S., 32, 161-169, (1978) · Zbl 0408.53033
[14] Darboux, G., Leçons sur la theorie des surfaces, Vol. 3, (1894), Gauthier-Villars Paris, 604/605
[15] Matsumoto, M., Projectively flat Finsler spaces of constant curvature, J. natl. acad. math., 1-2, 142-164, (1983), India · Zbl 0585.53020
[16] Rapcsák, A., Die bestimmung der grundfunktionen projektiv-ebener metrischer Räume, Publ. math., 9, 164-167, (1962), Debrecen · Zbl 0129.36004
[17] Hashiguchi, M.; Ichijyō, Y., Randers spaces with rectilinear geodesics, (), 33-40, (Math., Phys. and Chem.) · Zbl 0464.53024
[18] Hōjō, S., On geodesics of certain Finsler metrics, Tensor, N.S., 34, 211-217, (1980) · Zbl 0436.53023
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