Helicoidal surfaces satisfying \({\Delta ^{\mathrm{II}}\mathbf{G}=f(\mathbf{G}+C)}\). (English) Zbl 1362.53011
Summary: In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space \({\mathbb{R}^{3}}\), satisfying the condition \({\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}\), where \({\Delta ^{II}}\) is the Laplace operator with respect to the second fundamental form, \(f\) is a smooth function on the surface and \(C\) is a constant vector. Our main results state that helicoidal surfaces without parabolic points in \({\mathbb{R}^{3}}\) which satisfy the condition \({\Delta^{II} \mathbf{G}=f(\mathbf{G}+C)}\), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.
MSC:
| 53A05 | Surfaces in Euclidean and related spaces |
Keywords:
helicoidal surfaces; surfaces of revolution; Laplace operator; pointwise 1-type Gauss map; second fundamental formReferences:
| [1] | Baba-Hamed, Ch., Bekkar, M.: Helicoidal surfaces in the three-dimensional Lorentz-Minkowski space satisfying \[{\Delta^{II}r_i=\lambda_ir_i}\] ΔIIri=λiri. J. Geom. 100, 1-10 (2011) · Zbl 1244.53021 |
| [2] | Baba-Hamed, C., Bekkar, M.: Surfaces of revolution satisfying \[{\Delta^{II}\mathbf{G}=f(\mathbf{G}+C)}\] ΔIIG=f(G+C). Bull. Korean Math. Soc. 50(4), 1061-1067 (2013) · Zbl 1275.53006 |
| [3] | Baikoussis C., Blair D.E.: On the Gauss map of ruled surfaces. Glasg. Math. J. 34, 355-359 (1992) · Zbl 0762.53004 · doi:10.1017/S0017089500008946 |
| [4] | Baikoussis C., Chen B.-Y., Verstraelen L.: Ruled surfaces and tubes with finite type Gauss maps. Tokyo J. Math. 16, 341-349 (1993) · Zbl 0798.53055 · doi:10.3836/tjm/1270128488 |
| [5] | Chen B.-Y.: On submanifolds of finite type. Soochow J. Math. 9, 65-81 (1987) |
| [6] | Chen B.-Y.: Total Mean Curvature and Submanifolds of Finite Type. World Scientific Publishing, New Jersey (1984) · Zbl 0537.53049 · doi:10.1142/0065 |
| [7] | Chen B.-Y.: A report of submanifolds of finite type. Soochow J. Math. 22(2), 117-337 (1996) · Zbl 0867.53001 |
| [8] | Chen B.-Y., Choi M., Kim Y.H.: Surfaces of revolution with pointwise 1-type Gauss map. J. Korean Math. Soc. 42(3), 447-455 (2005) · Zbl 1082.53004 · doi:10.4134/JKMS.2005.42.3.447 |
| [9] | Chen B.-Y., Piccinni P.: Submanifolds with finite type Gauss map. Bull. Aust. Math. Soc. 35, 161-186 (1987) · Zbl 0672.53044 · doi:10.1017/S0004972700013162 |
| [10] | Choi M., Kim Y.H.: Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 38(4), 753-761 (2001) · Zbl 1003.53006 |
| [11] | Choi M., Kim D.-S., Kim Y.H.: Helicoidal surfaces with pointwise 1-type Gauss map. J. Korean Math. Soc. 46(1), 215-223 (2009) · Zbl 1159.53010 · doi:10.4134/JKMS.2009.46.1.215 |
| [12] | Kim Y.H., Yoon D.W.: Ruled surfaces with finite type Gauss map in Minkowski spaces. Soochow J. Math. 26, 85-96 (2000) · Zbl 0976.53020 |
| [13] | Kim Y.H., Yoon D.W.: Ruled surfaces with pointwise 1-type Gauss map. J. Geom. Phys. 34(3-4), 191-205 (2000) · Zbl 0962.53034 · doi:10.1016/S0393-0440(99)00063-7 |
| [14] | Kim Y.H., Yoon D.W.: On the Gauss map of ruled surfaces in Minkowski space. Rocky Mt. J. Math. 35(5), 1555-1581 (2005) · Zbl 1103.53010 · doi:10.1216/rmjm/1181069651 |
| [15] | Niang A.: Rotation surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 42(1), 23-27 (2005) · Zbl 1077.53006 · doi:10.4134/BKMS.2005.42.1.023 |
| [16] | Niang A.: On rotation surfaces in the Minkowski 3-dimensional space with pointwise 1-type Gauss map. J. Korean Math. Soc. 41(6), 1007-1021 (2004) · Zbl 1073.53029 · doi:10.4134/JKMS.2004.41.6.1007 |
| [17] | O’Neill B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983) · Zbl 0531.53051 |
| [18] | Senoussi, B., Bekkar, M.: Helicoidal surfaces in the three-dimensional Lorentz-Minkowski space satisfying \[{\Delta^{II}r=Ar}\] ΔIIr=Ar. Kyushu J. Math. 67, 327-338 (2013) · Zbl 1278.53025 |
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