Abstract
In this chapter, we begin with Minkowski normed spaces which appear as tangent spaces of Finsler manifolds, and recall Euler’s homogeneous function theorem as an important calculus tool throughout the book. Then we give the definition of a Finsler manifold, followed by some examples and a naturally induced (asymmetric) distance structure.
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References
Agrachev, A., Barilari, D., Rizzi, L.: Curvature: a variational approach. Mem. Am. Math. Soc. 256, 1225 (2018)
Álvarez Paiva, J.C., Durán, C.E.: Geometric invariants of fanning curves. Adv. Appl. Math. 42, 290–312 (2009)
Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. Kluwer Academic Publishers Group, Dordrecht (1993)
Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Differ. Geom. 66, 377–435 (2004)
Bucataru, I., Miron, R.: Finsler–Lagrange Geometry. Applications to Dynamical Systems. Editura Academiei Române, Bucharest (2007)
Foulon, P.: Géométrie des équations différentielles du second ordre. (French) Ann. Inst. H. Poincaré Phys. Théor. 45, 1–28 (1986)
Grifone, J.: Structure presque-tangente et connexions. I. (French) Ann. Inst. Fourier (Grenoble) 22, 287–334 (1972)
Katok, A.B.: Ergodic perturbations of degenerate integrable Hamiltonian systems. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 37, 539–576 (1973) English translation in Math. USSR-Izv. 7, 535–572 (1973)
Lee, P.W.Y.: Displacement interpolations from a Hamiltonian point of view. J. Funct. Anal. 265, 3163–3203 (2013)
Miron, R., Hrimiuc, D., Shimada, H., Sabau, S.V.: The Geometry of Hamilton and Lagrange Spaces. Kluwer Academic Publishers Group, Dordrecht (2001)
Ohta, S.: On the curvature and heat flow on Hamiltonian systems. Anal. Geom. Metr. Spaces 2, 81–114 (2014)
Randers, G.: On an asymmetrical metric in the fourspace of general relativity. Phys. Rev. (2) 59, 195–199 (1941)
Shen, Y.-B., Shen, Z.: Introduction to Modern Finsler Geometry. World Scientific Publishing Co., Singapore (2016)
Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Corrected reprint of the 1971 edition. Springer, New York, Berlin (1983)
Ziller, W.: Geometry of the Katok examples. Ergodic Theory Dynam. Syst. 3, 135–157 (1983)
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Ohta, Si. (2021). Finsler Manifolds. In: Comparison Finsler Geometry. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-80650-7_2
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DOI: https://doi.org/10.1007/978-3-030-80650-7_2
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