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Some remarks on Finsler manifolds with constant flag curvature

dc.contributor.author Bryant, Robert L
dc.date.accessioned 2016-08-25T14:03:12Z
dc.date.issued 2002
dc.identifier.uri https://hdl.handle.net/10161/12685
dc.description.abstract This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. 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. 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. 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.publisher UNIV HOUSTON
dc.relation.ispartof Houston Journal of Mathematics
dc.title Some remarks on Finsler manifolds with constant flag curvature
dc.type Journal article
duke.contributor.id Bryant, Robert L|0110365
pubs.begin-page 221
pubs.end-page 262
pubs.issue 2
pubs.organisational-group Duke
pubs.organisational-group Mathematics
pubs.organisational-group Trinity College of Arts & Sciences
pubs.publication-status Published
pubs.volume 28
duke.contributor.orcid Bryant, Robert L|0000-0002-4890-2471


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