Abstract
This chapter is devoted to the derivation of a natural notion of curvature via a Jacobi field, which is the variational vector field of a geodesic variation. This argument goes back to Ludwig Berwald’s important posthumous paper.
The appearance of a geodesic variation reminds us of a characterization of covariant derivatives by using the Riemannian metric g V associated with a vector field V whose integral curves are geodesics. In fact, this viewpoint leads us to a useful and inspiring description of the Finsler curvature as the Riemannian curvature of g V. The metric g V is also called an osculating Riemannian metric, and its application to the Riemannian characterization of the Finsler curvature goes back to Ottó Varga.
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Ohta, Si. (2021). Curvature. In: Comparison Finsler Geometry. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-80650-7_5
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DOI: https://doi.org/10.1007/978-3-030-80650-7_5
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