Isometric immersions of the hyperbolic space H n ( - 1 ) into H n + 1 ( - 1 )

Ze-Jun Hu

Colloquium Mathematicae (1999)

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

Abstract

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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.

How to cite

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Hu, 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

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  2. [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
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  8. [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. [9] A. M. Li, Spacelike hypersurfaces with constant Gauss-Kronecker curvature in Minkowski space, Arch. Math. (Basel) 64 (1995), 534-551. Zbl0828.53050
  10. [10] W. Massey, Spaces of Gaussian curvature zero in Euclidean 3 -space, Tôhoku Math. J. 14 (1962), 73-79. Zbl0114.36903
  11. [11] K. Nomizu, Isometric immersions of the hyperbolic plane into the hyperbolic space, Math. Ann. 205 (1973), 181-192. Zbl0256.53042
  12. [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. [13] B. O'Neill and E. Stiel, Isometric immersions of constant curvature manifolds, Michigan Math. J. 10 (1963), 335-339. Zbl0124.37402
  14. [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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