Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-19T15:58:09.569Z Has data issue: false hasContentIssue false
  • English
  • Français

On some 3-dimensional CR submanifolds in S 6

Published online by Cambridge University Press:  22 January 2016

Hideya Hashimoto  and
Katsuya Mashimo
Hideya Hashimoto
Affiliation:
Nippon Institute of Technology, 4-1, Gakuendai, Miyashiro, Minami-Saitama Gun, Saitama 345-8501, Japan, hideya@nit.ac.jp
Katsuya Mashimo
Affiliation:
Department of Mathematics, Tokyo University of Agriculture and Technology, Fuchu, Tokyo 183-0054, Japan, mashimo@cc.tuat.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give two types of 3-dimensional CR-submanifolds of the 6-dimensional sphere. First we study whether there exists a 3-dimensinal CR-submanifold which is obtained as an orbit of a 3-dimensional simple Lie subgroup of G 2. There exists a unique (up to G 2) 3-dimensional CR-submanifold which is obtained as an orbit of reducible representations of SU(2) on R 7. As orbits of the subgroup which corresponds to the irreducible representation of SU(2) on R 7, we obtained 2-parameter family of 3-dimensional CR-submanifolds. Next we give a generalization of the example which was obtained by K. Sekigawa.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 156 , 1999 , pp. 171 - 185
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

References

[1] Bryant, R. L., Submanifolds and special structures on the octonians, J. Diff. Geometry, 17 (1982), 185–232.Google Scholar
[2] Dadok, J., Polar coordinates induced by actions of compact Lie groups, Trans. A. M. S., 288 (1985), 125–137.CrossRefGoogle Scholar
[3] Dynkin, E.B., Semi-simple subalgebras of semi-simple Lie algebras, A.M.S.Transl. Ser. 2, 6 (1957), 111–244.Google Scholar
[4] Ejiri, N., Totally real submanifolds ina6-sphere, Proc. A. M. S., 83 (1981), 759–763.CrossRefGoogle Scholar
[5] Ejiri, N., Equivariant minimal immersions of S2 into S2m(1), Trans. A. M. S., 297 (1986), 105–124.Google Scholar
[6] Freudenthal, H., Oktaven, Ausnahmegruppen und Oktavengeometrie, Geometriae Dedicata, 19 (1985), 7–63.CrossRefGoogle Scholar
[7] Gray, A., Almost complex submanifolds of six sphere, Proc. A. M. S., 20 (1969), 277–279.CrossRefGoogle Scholar
[8] Harvey, R. and Lawson, H.B., Calibrated geometries, Acta Math., 148 (1982), 47–157.CrossRefGoogle Scholar
[9] Hsiung, W. Y. and Lawson, H. B., Minimal submanifolds of low cohomogenity, J. Diff. Geometry, 5 (1971), 1–38.Google Scholar
[10] Mal’cev, A.I., On semi-simple subgroups of Lie groups, A.M.S. Transl, Ser. 1, 9 (1950), 172–213.Google Scholar
[11] Mashimo, K., Homogeneous totally real submanifolds of S6 , Tsukuba J. Math., 9 (1985), 185–202.CrossRefGoogle Scholar
[12] Mashimo, K., Homogeneous CR submanifolds of P3(C), (in preparation).Google Scholar
[13] Sekigawa, K., Some CR-submanifolds in a 6-dimensioanl sphere, Tensor (N.S.), 6 (1984), 13–20.Google Scholar