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arXiv · 1512.06209

On a Class of Complete and Projectively Flat Finsler Metrics

Guojun Yang
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Abstract

An $(\alpha,\beta)$-manifold $(M,F)$ is a Finsler manifold with the Finsler metric $F$ being defined by a Riemannian metric $\alpha$ and $1$-form $\beta$ on the manifold $M$. In this paper, we classify $n$-dimensional $(\alpha,\beta)$-manifolds (non-Randers type) which are positively complete and locally projectively flat. We show that the non-trivial class is that $M$ is homeomorphic to the $n$-sphere $S^n$ and $(S^n,F)$ is projectively related to a standard spherical Riemannian manifold, and then we obtain some special geometric properties on the geodesics and scalar flag curvature of $F$ on $S^n$, especially when $F$ is a metric of general square type.

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Guojun Yang. 2015-12-19. On a Class of Complete and Projectively Flat Finsler Metrics. https://arxiv.org/abs/1512.06209

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