Isometric immersions from

the hyperbolic space into

Authors:
Hu Ze-Jun and Zhao Guo-Song

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2693-2697

MSC (1991):
Primary 53C42; Secondary 53C21

MathSciNet review:
1397002

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we transform the problem of determining isometric immersions from into 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.

**1.**Kinetsu Abe,*Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions*, Tôhoku Math. J. (2)**25**(1973), 425–444. MR**0350671****2.**Kinetsu Abe and Andrew Haas,*Isometric immersions of 𝐻ⁿ into 𝐻ⁿ⁺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****3.**Dirk Ferus,*Totally geodesic foliations*, Math. Ann.**188**(1970), 313–316. MR**0271872****4.**Dirk Ferus,*On isometric immersions between hyperbolic spaces*, Math. Ann.**205**(1973), 193–200. MR**0336665****5.**Philip Hartman and Louis Nirenberg,*On spherical image maps whose Jacobians do not change sign*, Amer. J. Math.**81**(1959), 901–920. MR**0126812****6.**Z. J. Hu and G. S. Zhao,*Classification of isometric immersions of the hyperbolic space into*, Geom. Dedicata (to appear).**7.**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**, 10.1007/BF01195136**8.**William S. Massey,*Surfaces of Gaussian curvature zero in Euclidean 3-space*, Tôhoku Math. J. (2)**14**(1962), 73–79. MR**0139088****9.**Katsumi Nomizu,*Isometric immersions of the hyperbolic plane into the hyperbolic space*, Math. Ann.**205**(1973), 181–192. MR**0336664****10.**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****11.**Barrett O’Neill and Edsel Stiel,*Isometric immersions of constant curvature manifolds*, Michigan Math. J.**10**(1963), 335–339. MR**0158329****12.**Bert G. Wachsmuth,*On the Dirichlet problem for the degenerate real Monge-Ampère equation*, Math. Z.**210**(1992), no. 1, 23–35. MR**1161168**, 10.1007/BF02571781

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
53C42,
53C21

Retrieve articles in all journals with MSC (1991): 53C42, 53C21

Additional 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

**Zhao Guo-Song**

Affiliation:
Department of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China

DOI:
https://doi.org/10.1090/S0002-9939-97-03905-1

Keywords:
Isometric immersion,
hyperbolic space,
Monge-Amp\`ere equation

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

Article copyright:
© Copyright 1997
American Mathematical Society