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Bryant settles a well-known open problem in Finsler geometry by proving that a reversible Finsler metric on the two-sphere with constant curvature is Riemannian. This paper can be downloaded from the arXiv.
A sphere theorem for non-reversible Finsler metrics
This paper extends Dazord's sphere theorem for reversible Finsler metrics and shows how to include the non-reversibility of the metric into the curvature estimates.
The link is to a preprint version of the paper that was published in
Mathematische Annalen |
This is an introduction to non-symmetric metrics from the point of view of metric geometry and variational calculus The preprint is available from the CVGMT server.
What is wrong with the Hausdorff measure in Finsler spaces
The authors show that if the Hausdorff measure is used as a notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily minimal, filling results such as that of Sergey Ivanov do not hold, and integral geometric formulas do not exist. On the other hand, using the Holmes-Thompson definition of volume, the authors prove a general Crofton formula for Finsler spaces and give a simple proof that their totally geodesic hypersurfaces are minimal. |