Isometric immersions of the hyperbolic space Hn(-1) into Hn+1(-1)
Hu, Ze-Jun
Colloquium Mathematicae, Tome 79 (1999), p. 17-23 / Harvested from The Polish Digital Mathematics Library
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We transform the problem of determining isometric immersions from Hn(-1) into Hn+1(-1) into that of solving equations of degenerate Monge-Ampère type on the unit ball Bn(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.

Publié le : 1999-01-01
EUDML-ID : urn:eudml:doc:210623 
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     title = {Isometric immersions of the hyperbolic space $H^n(-1)$ into $H^{n+1}(-1)$
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     journal = {Colloquium Mathematicae},
     volume = {79},
     year = {1999},
     pages = {17-23},
     zbl = {0915.53028},
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Hu, Ze-Jun. Isometric immersions of the hyperbolic space $H^n(-1)$ into $H^{n+1}(-1)$
            . Colloquium Mathematicae, Tome 79 (1999) pp. 17-23. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv79z1p17bwm/

[000] [1] K. Abe, Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions, Tôhoku Math. J. 25 (1973), 425-444. | Zbl 0283.53045

[001] [2] K. Abe and A. Haas, Isometric immersions of Hn into Hn+1, in: Proc. Sympos. Pure Math. 54, Part 3, Amer. Math. Soc., 1993, 23-30. | Zbl 0799.53061

[002] [3] K. Abe, H. Mori and H. Takahashi, A parametrization of isometric immersions between hyperbolic spaces, Geom. Dedicata 65 (1997), 31-46. | Zbl 0870.53044

[003] [4] D. Ferus, Totally geodesic foliations, Math. Ann. 188 (1970), 313-316. | Zbl 0194.52804

[004] [5] D. Ferus, On isometric immersions between hyperbolic spaces, ibid. 205 (1973), 193-200.

[005] [6] P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901-920. | Zbl 0094.16303

[006] [7] Z. J. Hu and G. S. Zhao, Classification of isometric immersions of the hyperbolic space H2 into H3, Geom. Dedicata 65 (1997), 47-57. | Zbl 0870.53045

[007] [8] Z. J. Hu and G. S. Zhao, Isometric immersions from the hyperbolic space H2(-1) into H3(-1), Proc. Amer. Math. Soc. 125 (1997), 2693-2697. | Zbl 0886.53043

[008] [9] A. M. Li, Spacelike hypersurfaces with constant Gauss-Kronecker curvature in Minkowski space, Arch. Math. (Basel) 64 (1995), 534-551. | Zbl 0828.53050

[009] [10] W. Massey, Spaces of Gaussian curvature zero in Euclidean 3-space, Tôhoku Math. J. 14 (1962), 73-79. | Zbl 0114.36903

[010] [11] K. Nomizu, Isometric immersions of the hyperbolic plane into the hyperbolic space, Math. Ann. 205 (1973), 181-192. | Zbl 0256.53042

[011] [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. | Zbl 0502.53038

[012] [13] B. O'Neill and E. Stiel, Isometric immersions of constant curvature manifolds, Michigan Math. J. 10 (1963), 335-339. | Zbl 0124.37402

[013] [14] B. G. Wachsmuth, On the Dirichlet problem for the degenerate real Monge-Ampère equation, Math. Z. 210 (1992), 23-35. | Zbl 0736.35050