Totally real submanifolds in satisfying Chen's equality
Authors:
Franki Dillen and Luc Vrancken
Journal:
Trans. Amer. Math. Soc. 348 (1996), 16331646
MSC (1991):
Primary 53B25; Secondary 53A10, 53B35, 53C25, 53C42
MathSciNet review:
1355070
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Abstract: In this paper, we study 3dimensional totally real submanifolds of . If this submanifold is contained in some 5dimensional totally geodesic , then we classify such submanifolds in terms of complex curves in lifted via the Hopf fibration . We also show that such submanifolds always satisfy Chen's equality, i.e. , where for every . Then we consider 3dimensional totally real submanifolds which are linearly full in and which satisfy Chen's equality. We classify such submanifolds as tubes of radius in the direction of the second normal space over an almost complex curve in .
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 [BVW1]
 J. Bolton, L. Vrancken and L.M. Woodward, On almost complex curves in the nearly Kähler 6sphere, Quart. J. Math. Oxford Ser. (2) 45 (1994), 407427. MR 95:07
 [BVW2]
 , Totally real minimal surfaces with noncircular ellipse of curvature in the nearly Kähler 6sphere, Proc. London Math. Soc. (to appear).
 [BW]
 J. Bolton, L.M. Woodward, Congruence theorems for harmonic maps from a Riemann surface into and , J. London Math. Soc. 45 (1992), 363376. MR 93k:58062
 [Br]
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 [CDVV1]
 B.Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Two equivariant totally real immersions into the nearly Kähler 6sphere and their characterization, Japanese J. Math. (N.S.) 21 (1995), 207222.
 [CDVV2]
 B.Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Characterizing a class of totally real submanifolds of by their sectional curvatures, Tôhoku Math. J. 47 (1995), 185198.
 [DVV1]
 F. Dillen, L. Verstraelen, L. Vrancken, On problems of U. Simon concerning minimal submanifolds of the nearly Kaehler 6sphere, Bull. Amer. Math. Soc. 19 (1988), 433438. MR 92b:53087
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 N. Ejiri, Totally real submanifolds in a 6sphere, Proc. Amer. Math. Soc. 83 (1981), 759763. MR 83a:53033
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Additional Information
Franki Dillen
Affiliation:
Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B3001 Leuven, Belgium
Email:
Franki.Dillen@wis.kuleuven.ac.be
Luc Vrancken
Affiliation:
Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B3001 Leuven, Belgium
Email:
Luc.Vrancken@wis.kuleuven.ac.be
DOI:
http://dx.doi.org/10.1090/S0002994796016261
PII:
S 00029947(96)016261
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
