On the Pontrjagin classes of spray manifolds
Zhongmin Shen, Runzhong Zhao
TL;DR
The paper shows that if a manifold $M$ admits a locally projectively flat spray, then all Pontrjagin classes vanish, i.e. $p_k=0$ for $k\ge1$, providing a topological obstruction to such sprays. The authors achieve this by a projective change using the $S$-curvature to convert a Douglas spray to a Berwald spray with $\hat{S}=0$, then demonstrate $\sigma_{2k}(\hat{\Omega})=0$ for all $k$, hence $(2\pi)^{2k}p_k(\xi)=0$. They further discuss concrete sphere constructions: on $\mathbf{S}^n$, locally projectively flat sprays arise as $G_P=G_0-2P y$ with homogeneous $P$, including Randers and Bryant families, and they verify smooth global realizations (e.g., via an ODE in the Bryant case). The results yield a clear topological constraint (notably excluding many complex projective spaces) and provide explicit models on spheres that realize the obstruction-free setting.
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.