There are no $\mathcal{C}^5$-Regular Pure $y$-Global Landsberg Surfaces

TL;DR

This paper proves that no pure $ ext{C}^5$ regular y-global Landsberg surfaces exist, using averaged connections and holonomy classification to show Landsberg implies Berwald in dimension two.

Contribution

It demonstrates the non-existence of pure $ ext{C}^5$ regular y-global Landsberg surfaces by linking Landsberg and Berwald conditions in two dimensions.

Findings

No pure $ ext{C}^5$ regular y-global Landsberg surfaces exist.

Landsberg condition implies Berwald condition in dimension two.

Classification of holonomies is used to establish the result.

Abstract

We show that there are not pure $\mathcal{C}^5$ regular y-global Landsberg surfaced. The proof is based on the averaged connection associated with the linear Chern's connection and the classification of irreducibles holonomies of torsion-free affine connections. The structure consists on exausting all the possible cases and showing that in dimension 2 Landsberg condition implies Berwald condition.