A rigidity theorem
for the Clifford tori in
Authors: Kazuyuki Enomoto, Yoshihisa Kitagawa and Joel L. Weiner
Journal: Proc. Amer. Math. Soc. 124 (1996), 265-268
MSC (1991): Primary 53C40; Secondary 53C45
MathSciNet review: 1285988
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Abstract: Let be the unit hypersphere in the 4-dimensional Euclidean space defined by . For each with , we denote by the Clifford torus in given by the equations and . The Clifford torus is a flat Riemannian manifold equipped with the metric induced by the inclusion map . In this note we prove the following rigidity theorem: If is an isometric embedding, then there exists an isometry of such that . We also show no flat torus with the intrinsic diameter is embeddable in except for a Clifford torus.
- Y. Kitagawa, Rigidity of the Clifford tori in , Math. Z. 198 (1988), 591--599. MR 0950585
- ------, Embedded flat tori in the unit 3-sphere J. Math. Soc. Japan 47 (1995), 275--296.
Affiliation: Faculty of Industrial Science and Technology, Science University of Tokyo, Oshamanbe, Hokkaido, 049-35 Japan
Affiliation: Department of Mathematics, Utsunomiya University, Mine-machi, Utsunomiya, 321 Japan
Joel L. Weiner
Affiliation: Department of Mathematics, University of Hawaii at Manoa, 2565 The Mall, Honolulu, Hawaii, 96822 U.S.A.
Keywords: Clifford torus, flat torus
Received by editor(s): December 16, 1993
Received by editor(s) in revised form: July 7, 1994
Communicated by: Christopher Croke
Article copyright: © Copyright 1996 American Mathematical Society