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Hypersurfaces of Type Number 2 in the Hyperbolic Four-Space and Their Extensions To Riemannian Geometry

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Non-Euclidean Geometries

Part of the book series: Mathematics and Its Applications ((MAIA,volume 581))

Abstract

Pseudo-symmetric spaces of constant type in dimension 3 are Riemannian three-manifolds whose Ricci tensor has, at all points, one double eigenvalue and one simple constant eigenvalue. In this paper we give a survey of our published results for the case when the constant Ricci eigenvalue is negative. In particular, we show that three-dimensional hypersurfaces of the hyperbolic space ℍ4 whose second fundamental form has rank 2 belong to this class. An explicit classification is presented in the case when the space admits so-called asymptotic foliation. Based on this, we show some existence theorems about local isometric embeddings of such spaces into ℍ4.

This research was supported by the grant GA ČR 201/02/0616 and was partly supported by the project MSM 113200007.

This research was supported by the Grant-in-Aid for Scientific Research (C) 14540066.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Oldřich Kowalski

  2. Tokyo Gakugei University, Koganei-shi Nukuikita-machi 4-1-1, Tokyo, 184-8501, Japan

    Masami Sekizawa

Authors
  1. Oldřich Kowalski
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  2. Masami Sekizawa
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Editor information

Editors and Affiliations

  1. Rutgers Center for Operations Research, Piscataway, New Jersey, USA

    András Prékopa

  2. Budapest University of Technology and Economics, Hungary

    Emil Molnár

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Kowalski, O., Sekizawa, M. (2006). Hypersurfaces of Type Number 2 in the Hyperbolic Four-Space and Their Extensions To Riemannian Geometry. In: Prékopa, A., Molnár, E. (eds) Non-Euclidean Geometries. Mathematics and Its Applications, vol 581. Springer, Boston, MA. https://doi.org/10.1007/0-387-29555-0_20

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