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COMPLETE REAL HYPERSURFACES AND SPECIAL 𝕂-LINE BUNDLES IN 𝕂-HYPERBOLIC SPACES

Department of Mathematics, Niigata University, Niigata 950-2181, Japan

Department of Information and Communication Engineering, Fukuoka Institute of Technology, Wajiro-Higashi, Fukuoka 811-0295, Japan

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KATSUHIRO SHIOHAMA

Department Applied Mathematics, Faculty of Sciences, Fukuoka University, 8-19-1 Nanakuma, Fukuoka 841-0180, Japan

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https://doi.org/10.1142/S0129167X13500821Cited by:1 (Source: Crossref)

Abstract

Using the geometry of geodesics, we discuss the global aspects of complete real hypersurfaces in hyperbolic spaces of constant holomorphic sectional curvature over any division algebra 𝕂. Our assumption is that the shape operator and the curvature transformation with respect to the normal unit have the same eigenspaces. Note that we do not assume constancy of the principal curvatures. Under this assumption, we give a complete global classification of such hypersurfaces. Since the argument is purely geometric, we need not vary the argument for different base algebras. The foliations of with totally geodesic leaves called 𝕂-lines play an important role.

Keywords:

AMSC: 53C40, 53C42, 53C20

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