Abstract
We show how to compute tensor derivatives and curvature tensors using affine connections. This allows for all computations to be obtained without using coordinate systems, in a way that parallels the computations appearing in classical Riemannian geometry. In particular, we obtain Bianchi identities for the curvature tensor of any anisotropic connection, we compare the curvature tensors of any two anisotropic connections, and we find a family of anisotropic connections which are well suited to study the geometry of Finsler metrics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bao, D., Chern, S.-S., Shen, Z.: An introduction to Riemann-Finsler geometry. Graduate Texts in Mathematics, vol. 200. Springer-Verlag, New York (2000)
Javaloyes, M.A.: Chern connection of a pseudo-Finsler metric as a family of affine connections. Publ. Math. Debrecen 84, 29–43 (2014)
Javaloyes, M.A.: Corrigendum to Chern connection of a pseudo-Finsler metric as a family of affine connections [mr3194771]. Publ. Math. Debrecen 85, 481–487 (2014)
Javaloyes, M.A.: Anisotropic tensor calculus. Int. J. Geom. Methods Mod. Phys. 16, suppl. 2, 1941001, 26 pp (2019)
Javaloyes, M.A., Soares, B.L.: Geodesics and Jacobi fields of pseudo-Finsler manifolds. Publ. Math. Debrecen 87, 57–78 (2015)
Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, (1996). Reprint of the 1963 original, A Wiley-Interscience Publication
Martínez, E., Cariñena, J.F., Sarlet, W.: Derivations of differential forms along the tangent bundle projection. Differ. Geom. Appl. 2, 17–43 (1992)
Martínez, E., Cariñena, J.F., Sarlet, W.: Derivations of differential forms along the tangent bundle projection. II. Differ. Geom. Appl. 3, 1–29 (1993)
Matthias, H.-H.: Zwei Verallgemeinerungen eines Satzes von Gromoll und Meyer, Bonner Mathematische Schriften [Bonn Mathematical Publications], 126, Universität Bonn, Mathematisches Institut, Bonn, 1980. Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn (1980)
O’Neill, B.: Semi-Riemannian geometry, vol. 103 of Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, (1983). With applications to relativity
Rund, H.: Direction-dependent connection and curvature forms. Abh. Math. Sem. Univ. Hamburg 50, 188–209 (1980)
Shen, Z.: Differential geometry of spray and Finsler spaces. Kluwer Academic Publishers, Dordrecht (2001)
Acknowledgements
The author warmly acknowledges useful discussions with Professors Amir Aazami (Clark University, USA) and Eduardo Martínez (University of Zaragoza, Spain), as well as some improvements suggested by the referees. The research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. It was partially supported by MINECO/FEDER project MTM2015-65430-P, MICINN/FEDER project PGC2018-097046-B-I00 and Fundación Séneca project 19901/GERM/15, Spain.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Javaloyes, M.Á. Curvature Computations in Finsler Geometry Using a Distinguished Class of Anisotropic Connections. Mediterr. J. Math. 17, 123 (2020). https://doi.org/10.1007/s00009-020-01560-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-020-01560-0