Some remarks on Finsler manifolds with constant flag curvature

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dc.creatorBryant, Robert L.
dc.date2001-07-31
dc.date.accessioned2026-07-07T04:42:48Z
dc.date.available2026-07-07T04:42:48Z
dc.descriptionThis article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. <p> The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in complex projective n-space. <p> The third remark is that there is a description of the Finsler metrics of constant curvature on the 2-sphere in terms of a Riemannian metric and 1-form on the space of its geodesics. In particular, this allows one to use any (Riemannian) Zoll metric of positive Gauss curvature on the 2-sphere to construct a global Finsler metric of constant positive curvature on the 2-sphere. <p> The fourth remark concerns the generality of the space of (local) Finsler metrics of constant positive flag curvature in dimension n+1>2 . It is shown that such metrics depend on n(n+1) arbitrary functions of n+1 variables and that such metrics naturally correspond to certain torsion-free S^1 x GL(n,R)-structures on 2n-manifolds. As a by-product, it is found that these groups do occur as the holonomy of torsion-free affine connections in dimension 2n, a hitherto unsuspected phenomenon.
dc.description32 pages, latex2e, amsart
dc.identifierhttps://arxiv.org/abs/math/0107228
dc.identifierhttp://arxiv.org/abs/math/0107228
dc.identifierHouston J. Math. 28 (2002), no. 2, 221--262.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/61940
dc.subjectDifferential Geometry
dc.subject53B40, 53C60, 58A15
dc.titleSome remarks on Finsler manifolds with constant flag curvature
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