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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Complete real hypersurfaces in compact rank one symmetric spaces


Authors: Tatsuyoshi Hamada and Katsuhiro Shiohama
Journal: Proc. Amer. Math. Soc. 137 (2009), 3905-3910
MSC (2000): Primary 53C20, 53C40
Published electronically: June 9, 2009
MathSciNet review: 2529899
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Abstract: The local classification of real hypersurfaces in a compact rank one symmetric space has been investigated by many people. Making use of the global behavior of geodesics on CROSS, we prove that a complete real hypersurface in a CROSS is a metric sphere if its shape operator and the curvature transformation with respect to the normal have the same eigenspaces at each point of it and if its principal curvatures are constant. We emphasize that our discussion is independent of the choice of the coefficient fields of projective spaces with constant holomorphic sectional curvature.


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Tatsuyoshi Hamada
Affiliation: Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, 8-19-1 Nanakuma, Fukuoka, 814-0180, Japan – and – Japan Science and Technology Agency, CREST, 5, Sanbancho, Chiyoda-ku, Tokyo, 102-0075, Japan
Email: hamada@holst.sm.fukuoka-u.ac.jp

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

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09959-6
PII: S 0002-9939(09)09959-6
Received by editor(s): October 15, 2008
Received by editor(s) in revised form: March 7, 2009
Published electronically: June 9, 2009
Additional Notes: The research of the first-named author was partially supported by Grant-in-Aid for Scientific Research (C), No. 18540104. The research of the second-named author was partially supported by Grant-in-Aid for Scientific Research (C), No. 19540107
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2009 American Mathematical Society