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Selecta Mathematica

, 3:161 | Cite as

Projectively flat Finsler 2-spheres of constant curvature

  • R. L. BryantEmail author
Article

Abstract.

After recalling the structure equations of Finsler structures on surfaces, I define a notion of "generalized Finsler structure" as a way of microlocalizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a surface and define a notion of "generalized path geometry" analogous to that of "generalized Finsler structure". I use these ideas to study the geometry of Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature K and whose geodesic path geometry is projectively flat, i.e., locally equivalent to that of straight lines in the plane. I show that, modulo diffeomorphism, there is a 2-parameter family of projectively flat Finsler structures on the sphere whose Finsler-Gauss curvature K is identically 1.

Keywords

Finsler geometry Projective geometry 

Copyright information

© Birkhäuser Verlag, Basel 1997

Authors and Affiliations

  1. 1.Department of Mathematics, Duke UniversityDurhamUSA

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