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A Characterization of Spheres in the Heisenberg Group H~n
Author: LinYiFan
Tutor: MaHui
School: Tsinghua University
Course: Mathematics
Keywords: subRiemannian the Heisenberg group spheres horizontal principal curvatures rotational invariant
CLC: O186.12
Type: Master's thesis
Year: 2013
Downloads: 3
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Abstract
The Heisenberg group is the simplest subRiemannian space. Spheres are kind ofmost important hypersurfaces in the Heisenberg group. In this paper we will give a characterization of spheres inH~n. Spheres in the subRiemannian Heisenberg groupH~narenot totally umbilical, they are a kind of spherical hypersurfaces with two distinct horizontal principal curvatures (move out two poles),λ1with multiplicity1andλ2=12λ1withmultiplicity2n2. If M is a C2immersed rotational invariant hypersurface inH~n, thenthe regular set of M has two distinct horizontal principal curvatures,λ1with multiplicity1andλ2with multiplicity2n2. Moreover, if M is connect andλ12=2λ1, then M iseither part of a sphere (whenλ10) inH~nor part of a Euclidean hyperplane orthogonal to the taxis (whenλ1=0). We extend the first variation formula for the horizontalareas of hypersurfaces, the Minkowski type formula and the horizontal geodesic equation in the Heisenberg group into higher dimension. We also proof that unlike inH1,inH~n(n≥2) the characteristic curve in rotational invariant hypersurfaces with constanthorizontal mean curvature may not satisfy the horizontal geodesic equation and give acounter example.

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CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Differential geometry,integral geometry > Differential Geometry > Riemannian geometry
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