Averaged Riemannian metrics and connections with application to locally conformal Berwald manifolds
The author recalls in his own formalism the notion of averaged Riemannian metric and averaged connection defined in [\textit{R. Gallego Torrome} and \textit{F. Etayo}, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 104, No. 1, 69--80 (2010; Zbl 1197.53023)]. Then he proves, among other, the following:NEWLINENEWLINENEWLINEi) If \((M,L)\) is a Landsberg manifold and \(g\) is the averaged metric for \((M,L)\), then the covariant derivative \(\nabla\) obtained from the Berwald derivative \(D\) of \((M,L)\) is the Levi-Civita derivation of \(g\).NEWLINENEWLINENEWLINEii) If \((M,L)\) is a locally conformal Berwald manifold and \(g\) is the averaged Riemannian metric for \((M,L)\), and the free-torsion covariant derivative \(\nabla\) satisfies a certain condition then \(\nabla\) is the Weyl connection for the conformal class of \(g\).
scientific article; zbMATH DE number 6083260