Abstract
In this chapter, we begin our study of differential calculus on Finsler manifolds. The main subject of the chapter is the geodesic equation as the Euler–Lagrange equation for the energy functional. To this end, some important quantities such as the fundamental and Cartan tensors are introduced. We will see that the metric definition of geodesics coincides with the variational definition as solutions to the geodesic equation. We also prove the Finsler analogue of the Hopf–Rinow theorem.
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References
Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann–Finsler Geometry. Springer, New York (2000)
Chavel, I.: Riemannian Geometry. A Modern Introduction, 2nd edn. Cambridge University Press, Cambridge (2006)
Shen, Z.: Lectures on Finsler Geometry. World Scientific Publishing Co., Singapore (2001)
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Ohta, Si. (2021). Properties of Geodesics. In: Comparison Finsler Geometry. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-80650-7_3
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DOI: https://doi.org/10.1007/978-3-030-80650-7_3
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Publisher Name: Springer, Cham
Print ISBN: 978-3-030-80649-1
Online ISBN: 978-3-030-80650-7
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