Higher order parallel surfaces in the Heisenberg group. (English) Zbl 1142.53017
The authors consider \(k\)-parallel surfaces, i.e. surfaces that satisfy \(\nabla^kh=0\), immersed in the Heisenberg group. They give a classification of such surfaces, proving that every connected \(k\)-parallel surface in the Heisenberg group \(Nil_3\) is an open part of a vertical cylinder on a plane curve, whose curvature function is a polynomial function of degree at most \(k-1\) of the arc length.
Reviewer: Marian Hotloś (Wrocław)
References:
| [1] | Backes, E.; Reckziegel, H., On symmetric submanifolds of spaces of constant curvature, Math. Ann., 263, 419-433 (1983) · Zbl 0499.53045 |
| [2] | Bekkar, M., Exemples de surfaces minimales dans l’espace de Heisenberg, Rend. Sem. Fac. Sci. Univ. Cagliari., 61, 123-130 (1991) · Zbl 0792.53056 |
| [3] | Belkhelfa, M.; Dillen, F.; Inoguchi, J., Surfaces with parallel second fundamental form in Bianchi-Cartan-Vranceanu spaces, Banach Center Publications, 57, 67-87 (2002) · Zbl 1029.53071 |
| [4] | Daniel, B., Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv., 82, 87-131 (2007), B. Daniel, preprint, 2005 · Zbl 1123.53029 |
| [5] | Dillen, F., The classification of hypersurfaces of a Euclidean space with parallel higher order fundamental form, Math. Z., 230, 635-643 (1990) · Zbl 0671.53007 |
| [6] | Dillen, F., Sur les hypersurfaces parallèles d’ordre supérieur, C. R. Acad. Sci. Paris, 311, 185-187 (1990) · Zbl 0702.53012 |
| [7] | Dillen, F., Hypersurfaces of a real space form with parallel higher order fundamental form, Soochow J. Math., 18, 321-338 (1992) · Zbl 0765.53013 |
| [8] | Inoguchi, J.; Kumamoto, T.; Ohsugi, N.; Suyama, Y., Differential geometry of curves and surfaces in 3-dimensional homogeneous spaces II, Fukuoka Univ. Sci. Reports, 30, 1, 17-47 (2000) · Zbl 0974.53039 |
| [9] | Inoguchi, J., Flat translation invariant surfaces in the 3-dimensional Heisenberg group, J. Geom., 82, 83-90 (2005) · Zbl 1082.53064 |
| [10] | Lawson, H. B., Local rigidity theorems for minimal hypersurfaces, Ann. of Math., 89, 187-197 (1969) · Zbl 0174.24901 |
| [11] | Lumiste, Ü., Submanifolds with a Van der Waerden-Bortolotti plane connection and parallelism of the third fundamental form, Izv. Vyssh. Uchebn. Mat., 31, 18-27 (1987) · Zbl 0616.53006 |
| [12] | Lumiste, Ü., Submanifolds with parallel fundamental form, (Handbook of Differential Geometry, vol. 1 (2000), Elsevier Science B.V.: Elsevier Science B.V. Amsterdam), 779-864 · Zbl 0964.53002 |
| [13] | Sanini, A., Gauss map of a surface of Heisenberg group, Bollettino U.M.I., 11B, 79-93 (1997) · Zbl 0886.53018 |
| [14] | Takeuchi, M., Parallel submanifolds in space forms, (Manifolds and Lie Groups—Papers in Honor of Yozo Matsusima (1981), Birkhäuser), 429-447 · Zbl 0481.53047 |
| [15] | J. Van der Veken, Higher order parallel surfaces in Bianchi-Cartan-Vranceanu spaces, Result. Math., in press; J. Van der Veken, Higher order parallel surfaces in Bianchi-Cartan-Vranceanu spaces, Result. Math., in press · Zbl 1156.53014 |
| [16] | Weisstein, E. W., Cornu spiral, From MathWorld—A Wolfram Web Resource |
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