In this paper the authors extend the theory of Riemannian submersions to Finsler spaces. As a consequence, they are able to construct Finsler metrics on complex and quaternionic projective spaces all of whose geodesics are geometrical circles. They also study the relation between isometric submersions and symplectic reduction, and show that Finsler submersions are flag-curvature non-increasing.
This paper appeared in
Proceedings of the Amer. Math. Soc.
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In the seventies, Schaeffer conjectured that the girth of a normed space --- the infimum of the lengths of all curves on the unit sphere that join a pair of antipodes --- equals the girth of its dual. This paper contains a simple proof using elementary symplectic and Finsler geometry. It is also shown that other results in convex geometry such as the equality of the Holmes-Thompson area of the unit sphere of a Minkowski space and that of its dual, or Crofton's formula for hypersurfaces in Minkowski spaces are also consequences of the same symplectic result that implies Schaeffer's conjecture.
This paper supersedes |
Roughly, the basic result of this paper is:
What measures to use, how to measure areas in these metric spaces,
how do integral geometry and symplectic geometry combine in the
proof? The paper can be downloaded from the web pages of
Electronic Research Announcements
In this paper, the authors give a remarkably simple formula for the k-area integrand of a Finsler manifolds in terms of the Fourier transform of the norms in each tangent space.
This paper appeared in the
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