Roughness of level sets of differentiable maps on Heisenberg group

Details Roughness of level sets of differentiable maps on Heisenberg group Roughness of level sets of differentiable maps on Heisenberg group

2011-10-17

arxiv.org/abs/1110.3634v1

Mathematics

Metric Geometry

in french, 54 pages, 7 figures

Scientific paper

We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $(\Bbb{H}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive analysis in a new case of a map $F\in C^1_H(\Bbb{H}^1, \Bbb{R}^2)$ with surjective horizontal differential (an analogue of the classical implicit function theorem). Among other results, we show that a level set of such map is locally a simple curve of Hausdorff sub-Riemannian dimension 2, but, surprisingly, in general its two-dimensional Hausdorff measure can be zero or infinity. Therefore, those level sets (called \textsf{vertical curves}) can be of rough nature and not belong to the class of intrinsic regular manifolds.

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