# Isometric immersions of the hyperbolic space ${H}^{n}(-1)$ into ${H}^{n+1}(-1)$

Colloquium Mathematicae (1999)

- Volume: 79, Issue: 1, page 17-23
- ISSN: 0010-1354

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topHu, Ze-Jun. "Isometric immersions of the hyperbolic space $H^n(-1)$ into $H^{n+1}(-1)$." Colloquium Mathematicae 79.1 (1999): 17-23. <http://eudml.org/doc/210623>.

@article{Hu1999,

abstract = {We transform the problem of determining isometric immersions from $H^n(-1)$ into $H^\{n+1\}(-1)$ into that of solving equations of degenerate Monge-Ampère type on the unit ball $B^n(1)$. By presenting one family of special solutions to the equations, we obtain a great many noncongruent examples of such isometric immersions with or without umbilic set.},

author = {Hu, Ze-Jun},

journal = {Colloquium Mathematicae},

keywords = {isometric immersion; Monge-Ampère type equation; hyperbolic space; Monge-Ampère equation},

language = {eng},

number = {1},

pages = {17-23},

title = {Isometric immersions of the hyperbolic space $H^n(-1)$ into $H^\{n+1\}(-1)$},

url = {http://eudml.org/doc/210623},

volume = {79},

year = {1999},

}

TY - JOUR

AU - Hu, Ze-Jun

TI - Isometric immersions of the hyperbolic space $H^n(-1)$ into $H^{n+1}(-1)$

JO - Colloquium Mathematicae

PY - 1999

VL - 79

IS - 1

SP - 17

EP - 23

AB - We transform the problem of determining isometric immersions from $H^n(-1)$ into $H^{n+1}(-1)$ into that of solving equations of degenerate Monge-Ampère type on the unit ball $B^n(1)$. By presenting one family of special solutions to the equations, we obtain a great many noncongruent examples of such isometric immersions with or without umbilic set.

LA - eng

KW - isometric immersion; Monge-Ampère type equation; hyperbolic space; Monge-Ampère equation

UR - http://eudml.org/doc/210623

ER -

## References

top- [1] K. Abe, Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions, Tôhoku Math. J. 25 (1973), 425-444. Zbl0283.53045
- [2] K. Abe and A. Haas, Isometric immersions of ${H}^{n}$ into ${H}^{n}+1$, in: Proc. Sympos. Pure Math. 54, Part 3, Amer. Math. Soc., 1993, 23-30. Zbl0799.53061
- [3] K. Abe, H. Mori and H. Takahashi, A parametrization of isometric immersions between hyperbolic spaces, Geom. Dedicata 65 (1997), 31-46. Zbl0870.53044
- [4] D. Ferus, Totally geodesic foliations, Math. Ann. 188 (1970), 313-316. Zbl0194.52804
- [5] D. Ferus, On isometric immersions between hyperbolic spaces, ibid. 205 (1973), 193-200.
- [6] P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901-920. Zbl0094.16303
- [7] Z. J. Hu and G. S. Zhao, Classification of isometric immersions of the hyperbolic space ${H}^{2}$ into ${H}^{3}$, Geom. Dedicata 65 (1997), 47-57. Zbl0870.53045
- [8] Z. J. Hu and G. S. Zhao, Isometric immersions from the hyperbolic space ${H}^{2}(-1)$ into ${H}^{3}(-1)$, Proc. Amer. Math. Soc. 125 (1997), 2693-2697. Zbl0886.53043
- [9] A. M. Li, Spacelike hypersurfaces with constant Gauss-Kronecker curvature in Minkowski space, Arch. Math. (Basel) 64 (1995), 534-551. Zbl0828.53050
- [10] W. Massey, Spaces of Gaussian curvature zero in Euclidean $3$-space, Tôhoku Math. J. 14 (1962), 73-79. Zbl0114.36903
- [11] K. Nomizu, Isometric immersions of the hyperbolic plane into the hyperbolic space, Math. Ann. 205 (1973), 181-192. Zbl0256.53042
- [12] V. 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. Zbl0502.53038
- [13] B. O'Neill and E. Stiel, Isometric immersions of constant curvature manifolds, Michigan Math. J. 10 (1963), 335-339. Zbl0124.37402
- [14] B. G. Wachsmuth, On the Dirichlet problem for the degenerate real Monge-Ampère equation, Math. Z. 210 (1992), 23-35. Zbl0736.35050

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