Isometric immersions from the hyperbolic space $H^2(-1)$ into $H^3(-1)$
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- by Hu Ze-Jun and Zhao Guo-Song
- Proc. Amer. Math. Soc. 125 (1997), 2693-2697
- DOI: https://doi.org/10.1090/S0002-9939-97-03905-1
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Abstract:
In this paper, we transform the problem of determining isometric immersions from $H^2(-1)$ into $H^3(-1)$ into that of solving a degenerate Monge-Ampère equation on the unit disc. By presenting one family of special solutions to the equation, we obtain a great many noncongruent examples of such isometric immersions with or without umbilic set.References
- Kinetsu Abe, Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions, Tohoku Math. J. (2) 25 (1973), 425–444. MR 350671, DOI 10.2748/tmj/1178241276
- Kinetsu Abe and Andrew Haas, Isometric immersions of $H^n$ into $H^{n+1}$, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 23–30. MR 1216607, DOI 10.1090/pspum/054.3/1216607
- Dirk Ferus, Totally geodesic foliations, Math. Ann. 188 (1970), 313–316. MR 271872, DOI 10.1007/BF01431465
- Dirk Ferus, On isometric immersions between hyperbolic spaces, Math. Ann. 205 (1973), 193–200. MR 336665, DOI 10.1007/BF01349229
- Philip Hartman and Louis Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901–920. MR 126812, DOI 10.2307/2372995
- Z. J. Hu and G. S. Zhao, Classification of isometric immersions of the hyperbolic space $H^2$ into $H^3$, Geom. Dedicata (to appear).
- An Min Li, Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space, Arch. Math. (Basel) 64 (1995), no. 6, 534–551. MR 1329827, DOI 10.1007/BF01195136
- William S. Massey, Surfaces of Gaussian curvature zero in Euclidean $3$-space, Tohoku Math. J. (2) 14 (1962), 73–79. MR 139088, DOI 10.2748/tmj/1178244205
- Katsumi Nomizu, Isometric immersions of the hyperbolic plane into the hyperbolic space, Math. Ann. 205 (1973), 181–192. MR 336664, DOI 10.1007/BF01349228
- V. I. Oliker and U. Simon, Codazzi tensors and equations of Monge-Ampère type on compact manifolds of constant sectional curvature, J. Reine Angew. Math. 342 (1983), 35–65. MR 703485
- Barrett O’Neill and Edsel Stiel, Isometric immersions of constant curvature manifolds, Michigan Math. J. 10 (1963), 335–339. MR 158329
- Bert G. Wachsmuth, On the Dirichlet problem for the degenerate real Monge-Ampère equation, Math. Z. 210 (1992), no. 1, 23–35. MR 1161168, DOI 10.1007/BF02571781
Bibliographic Information
- Hu Ze-Jun
- Affiliation: Department of Mathematics, Zhengzhou University, Zhengzhou, 450052, Henan, People’s Republic of China
- Address at time of publication: Department of Mathematics, Hangzhou University, Hangzhou, 310028, Zhejiang, People’s Republic of China
- MR Author ID: 346519
- ORCID: 0000-0003-2744-5803
- Zhao Guo-Song
- Affiliation: Department of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China
- Received by editor(s): January 12, 1996
- Received by editor(s) in revised form: April 12, 1996
- Additional Notes: This research was supported by the National Natural Science Foundation of China
- Communicated by: Christopher Croke
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2693-2697
- MSC (1991): Primary 53C42; Secondary 53C21
- DOI: https://doi.org/10.1090/S0002-9939-97-03905-1
- MathSciNet review: 1397002