Projectively flat Finsler 2-spheres of constant curvature

• R. L. Bryant ^[1]
  1. [1] Duke University

     Duke University

     Township of Durham, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 3, Nº. 2, 1997, págs. 161-203
• Idioma: inglés
• DOI: 10.1007/s000290050009
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• Resumen

  • 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.