Abstract
We study manifolds of the Finsler type whose tangent (pseudo-)Riemannian spaces are invariant under the (pseudo)orthogonal group. We construct the Cartan connection and study geodesics, extremals, and also motions. We establish that if the metric tensor of the space is a homogeneous tensor of the zeroth order with respect to the coordinates of the tangent vector, then the metric of the tangent space is realized on a cone of revolution. We describe the structure of geodesics on the cone as trajectories of motion of a free particle in a central field.
Similar content being viewed by others
References
I. E. Tamm, “On a curved momentum space,” in: Collected Scientific Works [in Russian], Vol. 2, Nauka, Moscow (1975), pp. 218–225.
I. E. Tamm and V. G. Vologodskii, “On the use of a curved momentum space for the construction of a nonlocal Euclidean field theory,” in: Collected Scientific Works [in Russian], Vol. 2, Nauka, Moscow (1975), pp. 226–253.
V. I. Panzhenskii and O. V. Sukhova, “Toward a geometry of spaces with a Tamm metric,” in: Collected Works of the Laptev International Geometry Seminar [in Russian] (Penza, 26–31 January 2004), Penza State Pedagogical Univ., Penza (2004), pp. 95–101.
H. Rund, The Differential Geometry of Finsler Spaces, Springer, Berlin (1959).
V. I. Pan’zhenskii, J. Math. Sci., 169, 297–314.
L. P. Eisenhart, Continuous Groups of Transformations, Princeton Univ. Press, Princeton, N. J. (1933).
V. F. Kagan, Subprojective Spaces [in Russian], Fizmatlit, Moscow (1961).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 189, No. 2, pp. 186–197, November, 2016.
Rights and permissions
About this article
Cite this article
Panzhenskii, V.I., Surina, O.P. Finsler generalization of the Tamm metric. Theor Math Phys 189, 1563–1573 (2016). https://doi.org/10.1134/S0040577916110039
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577916110039