On the two dimensional Berwald-Landsberg problem

TL;DR

This paper proves that in two dimensions, Landsberg surfaces with certain regularity conditions necessarily satisfy the Berwald condition, using holonomy representations of the averaged Chern connection.

Contribution

It demonstrates that all two-dimensional Landsberg surfaces with $ ext{C}^5$ regularity are actually Berwald surfaces, resolving a specific case of the Berwald-Landsberg problem.

Findings

No $ ext{C}^5$-regular pure Landsberg surfaces exist in two dimensions.

Landsberg condition implies Berwald condition in 2D.

Method involves holonomy representation analysis of the averaged Chern connection.

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.