On the Pontrjagin classes of spray manifolds

TL;DR

This paper proves that manifolds with locally projectively flat Finsler metrics or sprays have zero Pontrjagin classes, linking geometric flatness with topological invariants.

Contribution

It establishes that locally projectively flat spray manifolds necessarily have vanishing Pontrjagin classes, connecting geometric properties with topological invariants.

Findings

Pontrjagin classes vanish for locally projectively flat spray manifolds

The result applies to Finsler metrics and sprays in differential geometry

Provides a topological characterization of flat spray structures

Abstract

Locally projectively flat metrics (or sprays) form a rich class of metrics (or sprays) in Finsler and spray geometry. The characterization of such metrics is the Hilbert Fourth Problem in the regular case. In this paper we study the Pontrjagin classes of a manifold given a spray structure, and show that a manifold equipped with a locally projectively flat Finsler metric (or spray) has zero Pontrjagin classes.