Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 21, No. 2, pp. 199-204 (2005)

A new proof of Szabó's theorem on the Riemann-metrizability of Berwald manifolds

Cs. Vincze

University of Debrecen

Abstract: The starting point of the famous structure theorems on Berwald spaces due to Z.I. Szabo is an observation on the Riemann-metrizability of positive definite Berwald manifolds. It states that there always exists a Riemannian metric on the underlying manifold such that its Levi-Civita connection is just the canonical connection of the Berwald manifold. In this paper we present a new elementary proof of this theorem. After constructing a Riemannian metric by the help of integration of the canonical Riemann-Finsler metric on the indicatrix hypersurface it is proved that in case of Berwald manifolds the canonical connection and the Levi-Civita connection coincide.

Keywords: Finsler manifolds, Berwald manifolds, Riemann-metrizability

Classification (MSC2000): 53C60; 58B20

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