Isometric submersions of Finsler manifolds
by Juan Carlos Álvarez Paiva and Carlos E. Durán
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. 29 (2001), 2409--2417.
Reprints may be obtained from the authors:
J.C. Álvarez
C.E. Durán.
Dual spheres have the same girth
by Juan Carlos Álvarez Paiva
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 Holmes-Thompson area and symplectic geometry which the author has "retired from circulation".
Crofton formulas in projective Finsler spaces
by J. C. Álvarez Paiva and E. Fernandes
Roughly, the basic result of this paper is:
If for some metric d in Rn the length of a line segment is given by the measure of the set of hyperplanes intersecting it, then the area of a k-dimensional flat region is given by the measure of the set of (n-k)-flats intersecting it.
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 of the AMS at
Electronic Research Announcements
Fourier Transforms and the Holmes-Thompson volume
by J. C. Álvarez Paiva and E. Fernandes
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 International Mathematical Research Notices, vol. 19 (1999), 1031-1042.
Reprints are available from the authors:
J.C. Álvarez
E. Fernandes