Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-96-01626-1
MathSciNet review: 1355070
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $S^{6}(1)$.


References [Enhancements On Off] (What's this?)

  • [B] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, vol. 509, Springer, Berlin, 1976. MR 57:7444
  • [BPW] J. Bolton, F. Pedit and L.M. Woodward, Minimal surfaces and the affine Toda field model, J. Reine Angew. Math. 459 (1995), 119--150.
  • [BVW1] J. Bolton, L. Vrancken and L.M. Woodward, On almost complex curves in the nearly Kähler 6-sphere, Quart. J. Math. Oxford Ser. (2) 45 (1994), 407--427. MR 95:07
  • [BVW2] ------, Totally real minimal surfaces with non-circular ellipse of curvature in the nearly Kähler 6-sphere, Proc. London Math. Soc. (to appear).
  • [BW] J. Bolton, L.M. Woodward, Congruence theorems for harmonic maps from a Riemann surface into $\mathbb{C}P^{n}$ and $S^{n}$, J. London Math. Soc. 45 (1992), 363--376. MR 93k:58062
  • [Br] R.L. Bryant, Submanifolds and special structures on the octonians, J. Differential Geom. 17 (1982), 185--232. MR 84h:53091
  • [Ca1] E. Calabi, Minimal immersions of surfaces into Euclidean spheres, J. Differential Geom. 1 (1967), 111--125. MR 38:1616
  • [Ca2] E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc. 87 (1958), 407--438. MR 24:A558
  • [C] B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel) 60 (1993), 568--578. MR 94d:53093
  • [CDVV1] B.-Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Two equivariant totally real immersions into the nearly Kähler 6-sphere and their characterization, Japanese J. Math. (N.S.) 21 (1995), 207--222.
  • [CDVV2] B.Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Characterizing a class of totally real submanifolds of $S^{6}(1)$ by their sectional curvatures, Tôhoku Math. J. 47 (1995), 185--198.
  • [DVV1] F. Dillen, L. Verstraelen, L. Vrancken, On problems of U. Simon concerning minimal submanifolds of the nearly Kaehler 6-sphere, Bull. Amer. Math. Soc. 19 (1988), 433--438. MR 92b:53087
  • [DVV2] F. Dillen, L. Verstraelen and L. Vrancken, Classification of totally real 3-dimensional submanifolds of $S^{6}(1)$ with $K \ge \tfrac{1}{16}$, J. Math. Soc. Japan 42 (1990), 565--584. MR 91k:53064
  • [DV] F. Dillen, L. Vrancken, C-totally real submanifolds of Sasakian space forms, J. Math. Pures Appl.(9) 69 (1990), 85--93. MR 91d:53077
  • [E1] N. Ejiri, Totally real submanifolds in a 6-sphere, Proc. Amer. Math. Soc. 83 (1981), 759--763. MR 83a:53033
  • [E2] N. Ejiri, Equivariant minimal immersions of $S^{2}$ into $S^{2m}$, Trans. Amer. Math. Soc. 297 (1986), 105--124. MR 87k:58061
  • [Er] J. Erbacher, Reduction of the codimension of an isometric immersions, J. Differential Geom. 5 (1971), 333--340. MR 44:5897
  • [HL] R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47--157. MR 85i:53058
  • [M] K. Mashimo, Homogeneous totally real submanifolds of $S^{6}(1)$, Tsukuba J. Math. 9 (1985), 185--202. MR 86j:53083
  • [S] K. Sekigawa, Almost complex submanifolds of a 6-dimensional sphere, K\={o}dai Math. J. 6 (1983), 174--185. MR 84i:53059
  • [Sp] M. Spivak, A comprehensive introduction to Differential Geometry, Vol.1, Publish or Perish, Houston, 1970. MR 42:6726
  • [W] R.M.W. Wood, Framing the exceptional Lie group $G_{2}$, Topology 15 (1976), 303--320. MR 58:7665
  • [YI] K. Yano, S. Ishihara, Invariant submanifolds of an almost contact manifold, K\B{o}dai Math. Sem. Rep. 21 (1969), 350--364. MR 40:1946

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53B25, 53A10, 53B35, 53C25, 53C42

Retrieve articles in all journals with MSC (1991): 53B25, 53A10, 53B35, 53C25, 53C42


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: https://doi.org/10.1090/S0002-9947-96-01626-1
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

American Mathematical Society