On the Berwald-Landsberg problem

Details On the Berwald-Landsberg problem On the Berwald-Landsberg problem

2011-10-25

arxiv.org/abs/1110.5680v3

Mathematics

Differential Geometry

24 pages, Minor changes; misprints corrected

Scientific paper

Given a Finsler manifold $({\bf M},F)$, one can define natural {\it averaged} Riemannian manifolds living on ${\bf M}$ by averaging on the indicatrix ${\bf I}_x$ the fundamental tensor $g$. In this paper we determine the Levi-Civita connection for such averaged Riemannian manifolds. We apply the result to the case when $({\bf M},F)$ is a Landsberg space. Given a particular averaging procedure, the invariance of the averaged metric under certain homotopy in the space of Finsler manifolds over {\bf M} is shown. Using such result we prove that any Landsberg space which is of class $\mathcal{C}^4$ is a Berwald space.

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