Bulletin of the Korean Mathematical Society (대한수학회보)

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HYPERSURFACES IN 𝕊4 THAT ARE OF Lk-2-TYPE

  • Received : 2015.05.27
  • Published : 2016.05.31
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Abstract

In this paper we begin the study of $L_k$-2-type hypersurfaces of a hypersphere ${\mathbb{S}}^{n+1}{\subset}{\mathbb{R}}^{n+2}$ for $k{\geq}1$ Let ${\psi}:M^3{\rightarrow}{\mathbb{S}}^4$ be an orientable $H_k$-hypersurface, which is not an open portion of a hypersphere. Then $M^3$ is of $L_k$-2-type if and only if $M^3$ is a Clifford tori ${\mathbb{S}}^1(r_1){\times}{\mathbb{S}}^2(r_2)$, $r^2_1+r^2_2=1$, for appropriate radii, or a tube $T^r(V^2)$ of appropriate constant radius r around the Veronese embedding of the real projective plane ${\mathbb{R}}P^2({\sqrt{3}})$.

Keywords

Acknowledgement

Supported by : MINECO (Ministerio de Economia y Competitividad), FEDER (Fondo Europeo de Desarrollo Regional)

References

  1. L. J. Alias, A. Ferrandez, and P. Lucas, Surfaces in the 3-dimensional LorentzMinkowski space satisfying ${\Delta}$$_{x}$ = A$_{x}$ + B, Pacific J. Math. 156 (1992), no. 2, 201-208. https://doi.org/10.2140/pjm.1992.156.201
  2. L. J. Alias, A. Ferrandez, and P. Lucas, Submanifolds in pseudo-Euclidean spaces satisfying the condition ${\Delta}x$ = Ax+B, Geom. Dedicata 42 (1992), no. 3, 345-354. https://doi.org/10.1007/BF02414072
  3. L. J. Alias, A. Ferrandez, and P. Lucas, Hypersurfaces in space forms satisfying the condition ${\Delta}x$ = Ax + B, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1793-1801. https://doi.org/10.1090/S0002-9947-1995-1257095-8
  4. L. J. Alias and N. Gurbuz, An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (2006), 113-127.
  5. L. J. Alias and M. B. Kashani, Hypersurfaces in space forms satisfying the condition $L_k{\Psi}\;=\;A{\Psi}+b$, Taiwanese J. Math. 14, no. 5 (2010), no. 5, 1957-1978. https://doi.org/10.11650/twjm/1500406026
  6. M. Barros and O. J. Garay, 2-type surfaces in $\mathbb{S}^3$, Geom. Dedicata 24 (1987), no. 3, 329-336. https://doi.org/10.1007/BF00181605
  7. E. Cartan, Familles de surfaces isoparametriques dans les espaces a courbure constante, Ann. Mat. Pura Appl. 17 (1938), no. 1, 177-191. https://doi.org/10.1007/BF02410700
  8. E. Cartan, Sur des familles remarquables d'hypersurfaces isoparametriques dans les espaces spheriques, Math. Z. 45 (1939), 335-367. https://doi.org/10.1007/BF01580289
  9. E. Cartan, Sur quelque familles remarquables d'hypersurfaces, C. R. Congres Math. Liege (1939), 30-41.
  10. S. Chang, A closed hypersurface of constant scalar curvature and constant mean curvature in $S^{4}$ is isoparametric, Comm. Anal. Geom. 1 (1993), 71-100. https://doi.org/10.4310/CAG.1993.v1.n1.a4
  11. S. Chang, On closed hypersurfaces of constant scalar curvatures and mean curvatures in $S^{n+1}$, Pacific J. Math. 165 (1994), no. 1, 67-76. https://doi.org/10.2140/pjm.1994.165.67
  12. B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 1. World Scientific Publishing Co., Singapore, 1984.
  13. B. Y. Chen, Finite Type Submanifolds and Generalizations, University of Rome, Rome, 1985.
  14. B. Y. Chen, Finite type submanifolds in pseudo-Euclidean spaces and applications, Kodai Math. J. 8 (1985), no. 3, 358-375. https://doi.org/10.2996/kmj/1138037104
  15. B. Y. Chen, 2-type submanifolds and their applications, Chinese J. Math. 14 (1986), no. 1, 1-14.
  16. B. Y. Chen, Tubular hypersurfaces satisfying a basic equality, Soochow J. Math. 20 (1994), no. 4, 569-586.
  17. B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (1996), no. 2, 117-337.
  18. B. Y. Chen, Some open problems and conjectures on submanifolds of finite type: recent development, Tamkang J. Math. 45 (2014), no. 1, 87-108. https://doi.org/10.5556/j.tkjm.45.2014.1564
  19. B. Y. Chen, M. Barros, and O. J. Garay, Spherical finite type hypersurfaces, Alg. Groups Geom. 4 (1987), no. 1, 58-72.
  20. B. Y. Chen and M. Petrovic, On spectral decomposition of immersions of finite type, Bull. Austral. Math. Soc. 44 (1991), no. 1, 117-129. https://doi.org/10.1017/S0004972700029518
  21. S. De Almeida and F. Brito, Closed 3-dimensional hypersurfaces with constant mean curvature and constant scalar curvature, Duke Math. J. 61 (1990), no. 1, 195-206. https://doi.org/10.1215/S0012-7094-90-06109-5
  22. F. Dillen, J. Pas, and L. Verstraelen, On surfaces of finite type in Euclidean 3-space, Kodai Math. J. 13 (1990), no. 1, 10-21. https://doi.org/10.2996/kmj/1138039155
  23. V. N. Faddeeva, Computational Methods of Linear Algebra, Dover Publ. Inc, 1959, New York.
  24. O. J. Garay, An extension of Takahashi's theorem, Geom. Dedicata 34 (1990), no. 2, 105-112. https://doi.org/10.1007/BF00147319
  25. T. Hasanis and T. Vlachos, A local classification of 2-type surfaces in $S^{3}$, Proc. Amer. Math. Soc. 112, no. 2 (1991), 533-538. https://doi.org/10.1090/S0002-9939-1991-1059626-1
  26. T. Hasanis and T. Vlachos, Spherical 2-type hypersurfaces, J. Geometry 40 (1991), no. 1-2, 82-94. https://doi.org/10.1007/BF01225875
  27. T. Hasanis and T. Vlachos, Hypersurfaces of $E^{n+1}$ satisfying ${\Delta}x$ = Ax + B, J. Austral. Math. Soc. Ser. A 53 (1992), no. 3, 377-384. https://doi.org/10.1017/S1446788700036545
  28. S. M. B. Kashani, On some $L_{1}$-finite type (hyper)surfaces in $\mathbb{R}^{n+1}$, Bull. Korean Math. Soc. 46 (2009), no. 1, 35-43. https://doi.org/10.4134/BKMS.2009.46.1.035
  29. U. J. J. Leverrier, Sur les variations seculaires des elements elliptiques des sept plan'etes principales, J. de Math. s.1 5 (1840), 220-254.
  30. P. Lucas and H. F. Ramirez-Ospina, Hypersurfaces in the Lorentz-Minkowski space satisfying $L_k{\Psi}\;=\;A{\Psi}+b$, Geom. Dedicata 153 (2011), 151-175. https://doi.org/10.1007/s10711-010-9562-z
  31. P. Lucas and H. F. Ramirez-Ospina, Hypersurfaces in non-flat Lorentzian space forms satisfying $L_k{\Psi}\;=\;A{\Psi}+b$, Taiwanese J. Math. 16 (2012), no. 3, 1173-1203. https://doi.org/10.11650/twjm/1500406685
  32. P. Lucas and H. F. Ramirez-Ospina, Hypersurfaces in pseudo-Euclidean spaces satisfying a linear condition on the linearized operator of a higher order mean curvature, Differential Geom. Appl. 31 (2013), no. 2, 175-189. https://doi.org/10.1016/j.difgeo.2013.01.002
  33. P. Lucas and H. F. Ramirez-Ospina, Hypersurfaces in non-flat pseudo-Riemannian space forms satisfying a linear condition in the linearized operator of a higher order mean curvature, Taiwanese J. Math. 17 (2013), no. 1, 15-45. https://doi.org/10.11650/tjm.17.2013.1738
  34. M. A. Magid, Lorentzian isoparametric hypersurfaces, Pacific J. Math. 118 (1985), no. 1, 165-197. https://doi.org/10.2140/pjm.1985.118.165
  35. A. Mohammadpouri and S. M. B. Kashani, On some $L_{k}$-finite-type Euclidean hypersurfaces, ISRN Geometry 2012 (2012), article ID 591296, 23 pages.
  36. H. Munzner, Isoparametrische hyperflachen in spharen. I and II, Math. Ann. 251 (1980), no. 1, 57-71 https://doi.org/10.1007/BF01420281
  37. H. Munzner, Isoparametrische hyperflachen in spharen. I and II, Math. Ann. 256 (1981), no. 2, 215-232. https://doi.org/10.1007/BF01450799
  38. B. O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, New York London, 1983.
  39. J. Park, Hypersurfaces satisfying the equation ${\Delta}x$ = Rx+ b, Proc. Amer. Math. Soc. 120 (1994), no. 1, 317-328. https://doi.org/10.1090/S0002-9939-1994-1189750-7
  40. R. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geom. 8 (1973), 465-477. https://doi.org/10.4310/jdg/1214431802