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Geometric meanings of curvatures in Finsler geometry. (English) Zbl 0991.53051
Slovák, Jan (ed.) et al., The proceedings of the 20th winter school “Geometry and physics”, Srní, Czech Republic, January 15-22, 2000. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 66, 165-178 (2001).
In Finsler geometry, we use calculus to study the geometry of regular inner metric spaces. In this paper the author briefly discusses various curvatures and their geometric meanings from the metric geometry point of view, without going into the forest of tensors. The first four sections are devoted to the consideration of all quantities which vanish in Riemannian geometry. The fifth section is concerned with the Riemann curvature which was extended by Berwald (1926) to the Finslerian case. Then some well-known results due to S. Numata (1975), T. Okada (1983), H. Akbar-Zadeh (1988), R. Bryant (1996; 1997) and the present author on Finsler spaces of scalar curvature and of constant curvature are discussed.
For the entire collection see [Zbl 0961.00020].

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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