On a Class of Complete and Projectively Flat Finsler Metrics

2015/12/19 by Yang, Guojun
#53A20 #53C60 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1512.06209

Abstract

An (α,β)-manifold (M,F) is a Finsler manifold with the Finsler metric F being defined by a Riemannian metric α and 1-form β on the manifold M. In this paper, we classify n-dimensional (α,β)-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 Sⁿ and (Sⁿ,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 Sⁿ, especially when F is a metric of general square type.

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