On the two dimensional Berwald-Landsberg problem

Abstract

The Berwald-Landsberg problem is considered for two dimensional manifolds. We sketch the proof that there are not ${\mathcal{C}}^5$-regular $y$-global pure Landsberg surfaces. The method used consists on considerer the holonomy representation of the averaged Chern connection and then exhausting all the possible cases, showing that in dimension 2 Landsberg condition implies Berwald condition.

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