Cheng-Yau operator and Gauss map of surfaces of revolution. (English) Zbl 1356.53005
Summary: We study the Gauss map \(G\) of surfaces of revolution in the 3-dimensional Euclidean space \({{\mathbb {E}}^3}\) with respect to the so-called Cheng-Yau operator \(\square \) acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss map \(G\) satisfying \(\square G=AG\) for some \(3\times 3\) matrix \(A\) are the planes, right circular cones, circular cylinders, and spheres.
References:
| [1] | Alias, Luis J., Gurbuz, N.: An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures. Geom. Dedic. 121, 113-127 (2006) · Zbl 1119.53004 · doi:10.1007/s10711-006-9093-9 |
| [2] | Baikoussis, C., Blair, D.E.: On the Gauss map of ruled surfaces. Glasgow Math. J. 34(3), 355-359 (1992) · Zbl 0762.53004 · doi:10.1017/S0017089500008946 |
| [3] | Baikoussis, C., Verstraelen, L.: On the Gauss map of helicoidal surfaces. Rend. Sem. Mat. Messina Ser. II 2(16), 31-42 (1993) · Zbl 0859.53001 |
| [4] | Chen, B.-Y., Piccinni, P.: Submanifolds with finite type Gauss map. Bull. Austral. Math. Soc. 35(2), 161-186 (1987) · Zbl 0672.53044 · doi:10.1017/S0004972700013162 |
| [5] | Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225(3), 195-204 (1977) · Zbl 0349.53041 · doi:10.1007/BF01425237 |
| [6] | Choi, S.M.: On the Gauss map of surfaces of revolution in a 3-dimensional Minkowski space. Tsukuba J. Math. 19(2), 351-367 (1995) · Zbl 0853.53014 |
| [7] | Choi, S.M.: On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space. Tsukuba J. Math. 19(2), 285-304 (1995) · Zbl 0855.53010 |
| [8] | Choi, S.M., Kim, D.-S., Kim, Y.H., Yoon, D.W.: Circular cone and its Gauss map. Colloq. Math. 129(2), 203-210 (2012) · Zbl 1268.53002 · doi:10.4064/cm129-2-4 |
| [9] | Dillen, F., Pas, J., Verstraelen, L.: On the Gauss map of surfaces of revolution. Bull. Inst. Math. Acad. Sin. 18(3), 239-246 (1990) · Zbl 0727.53005 |
| [10] | do Carmo, Manfredo P.: Differential geometry of curves and surfaces. Translated from the Portuguese, Prentice-Hall Inc, Englewood Cliffs (1976) · Zbl 0326.53001 |
| [11] | Dursun, U.: Flat surfaces in the Euclidean space \[E^3\] E3 with pointwise 1-type Gauss map. Bull. Malays. Math. Sci. Soc. (2) 33(3), 469-478 (2010) · Zbl 1203.53003 |
| [12] | Ki, U.-H., Kim, D.-S., Kim, Y.H., Roh, Y.-M.: Surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space. Taiwan. J. Math. 13(1), 317-338 (2009) · Zbl 1178.35018 |
| [13] | Kim, D.-S.: On the Gauss map of quadric hypersurfaces. J. Korean Math. Soc. 31(3), 429-437 (1994) · Zbl 0818.53007 |
| [14] | Kim, D.-S.: On the Gauss map of hypersurfaces in the space form. J. Korean Math. Soc. 32(3), 509-518 (1995) · Zbl 0844.53004 |
| [15] | Kim, D.-S., Kim, Y.H.: Surfaces with planar lines of curvature. Honam Math. J. 32, 777-790 (2010) · Zbl 1215.53007 · doi:10.5831/HMJ.2010.32.4.777 |
| [16] | Kim, D.-S., Kim, Y.H., Yoon, D.W.: Extended B-scrolls and their Gauss maps. Indian J. Pure Appl. Math. 33(7), 1031-1040 (2002) · Zbl 1019.53004 |
| [17] | Kim, D.-S., Song, B.: On the Gauss map of generalized slant cylindrical surfaces. J. Korea Soc. Math. Educ. Ser. B 20(3), 149-158 (2013) · Zbl 1295.53004 |
| [18] | Kim, Y.H., Turgay, N.C.: Surfaces in \[E^3\] E3 with \[L_1\] L1-pointwise 1-type Gauss map. Bull. Korean Math. Soc. 50(3), 935-949 (2013) · Zbl 1270.53003 · doi:10.4134/BKMS.2013.50.3.935 |
| [19] | 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 |
| [20] | Levi-Civita, T.: Famiglie di superficie isoparametriche nellordinario spacio euclideo. Rend. Acad. Lincei 26, 355-362 (1937) · Zbl 0018.08702 |
| [21] | Ruh, E.A., Vilms, J.: The tension field of the Gauss map. Trans. Am. Math. Soc. 149, 569-573 (1970) · Zbl 0199.56102 · doi:10.1090/S0002-9947-1970-0259768-5 |
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