What is wrong with the Hausdorff measure in Finsler spaces

Details What is wrong with the Hausdorff measure in Finsler spaces What is wrong with the Hausdorff measure in Finsler spaces

2004-08-30

arxiv.org/abs/math/0408413v1

Adv. Math. 204 (2006), no. 2, 647-663

Mathematics

Differential Geometry

16 pages

Scientific paper

We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily minimal, filling results such as those of S. Ivanov do not hold, and integral-geometric formulas do not exist. On the other hand, using the Holmes-Thompson definition of volume, we prove a general Crofton formula for Finsler spaces and give an easy proof that their totally geodesic hypersurfaces are minimal.

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