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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Totally real submanifolds in $S^{6}(1)$
satisfying Chen's equality


Authors: Franki Dillen and Luc Vrancken
Journal: Trans. Amer. Math. Soc. 348 (1996), 1633-1646
MSC (1991): Primary 53B25; Secondary 53A10, 53B35, 53C25, 53C42
MathSciNet review: 1355070
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Abstract: In this paper, we study 3-dimensional totally real submanifolds of $S^{6}(1)$. If this submanifold is contained in some 5-dimensional totally geodesic $S^{5}(1)$, then we classify such submanifolds in terms of complex curves in $\mathbb{C}P^{2}(4)$ lifted via the Hopf fibration $S^{5}(1)\to \mathbb{C}P^{2}(4)$. We also show that such submanifolds always satisfy Chen's equality, i.e. $\delta _{M} = 2$, where $\delta _{M}(p)=\tau (p)-\inf K(p)$ for every $p\in M$. Then we consider 3-dimensional totally real submanifolds which are linearly full in $S^{6}(1)$ and which satisfy Chen's equality. We classify such submanifolds as tubes of radius $\pi /2$ in the direction of the second normal space over an almost complex curve in $S^{6}(1)$.


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Additional Information

Franki Dillen
Affiliation: Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B-3001 Leuven, Belgium
Email: Franki.Dillen@wis.kuleuven.ac.be

Luc Vrancken
Affiliation: Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B-3001 Leuven, Belgium
Email: Luc.Vrancken@wis.kuleuven.ac.be

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01626-1
PII: S 0002-9947(96)01626-1
Received by editor(s): April 19, 1995
Additional Notes: The authors are Senior Research Assistants of the National Fund for Scientific Research (Belgium).
The authors would like to thank J. Bolton and L.M. Woodward for helpful discussions.
Article copyright: © Copyright 1996 American Mathematical Society