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A Characterization of Spheres in the Heisenberg Group H~n

Author: LinYiFan
Tutor: MaHui
School: Tsinghua University
Course: Mathematics
Keywords: sub-Riemannian the Heisenberg group spheres horizontal principal cur-vatures rotational invariant
CLC: O186.12
Type: Master's thesis
Year: 2013
Downloads: 3
Quote: 0
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Abstract


The Heisenberg group is the simplest sub-Riemannian space. Spheres are kind ofmost important hypersurfaces in the Heisenberg group. In this paper we will give a char-acterization of spheres inH~n. Spheres in the sub-Riemannian Heisenberg groupH~narenot totally umbilical, they are a kind of spherical hypersurfaces with two distinct horizon-tal 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 orthogo-nal to the t-axis (whenλ1=0). We extend the first variation formula for the horizontalareas of hypersurfaces, the Minkowski type formula and the horizontal geodesic equa-tion 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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