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

 

A rigidity theorem
for the Clifford tori in $S^3$


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 $S^3$ be the unit hypersphere in the 4-dimensional Euclidean space $\mathbb R^4$ defined by $\sum_{i=1}^4 x_i^2=1$. For each $\theta$ with $0<\theta<\pi/2$, we denote by $M_\theta$ the Clifford torus in $S^3$ given by the equations $x_1^2+x_2^2=\cos^2\theta$ and $x_3^2+x_4^2= \sin^2\theta$. The Clifford torus $M_\theta$ is a flat Riemannian manifold equipped with the metric induced by the inclusion map $i_\theta\colon M_\theta\to S^3$. In this note we prove the following rigidity theorem: If $f\colon M_\theta \to S^3$ is an isometric embedding, then there exists an isometry $A$ of $S^3$ such that $f=A\circ i_\theta$. We also show no flat torus with the intrinsic diameter $\le\pi$ is embeddable in $S^3$ except for a Clifford torus.


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Additional Information

Kazuyuki Enomoto
Affiliation: Faculty of Industrial Science and Technology, Science University of Tokyo, Oshamanbe, Hokkaido, 049-35 Japan
Email: enomoto@it.osha.sut.ac.jp

Yoshihisa Kitagawa
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.
Email: joel@math.hawaii.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03001-8
PII: S 0002-9939(96)03001-8
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