Nagoya Mathematical Journal

On some 3-dimensional CR submanifolds in $S\sp 6$

Hideya Hashimoto and Katsuya Mashimo

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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 ${\bf R}^7$. As orbits of the subgroup which corresponds to the irreducible representation of $SU(2)$ on ${\bf R}^7$, we obtained 2-parameter family of 3-dimensional CR-submanifolds. Next we give a generalization of the example which was obtained by K. Sekigawa.

Article information

Nagoya Math. J., Volume 156 (1999), 171-185.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)


Hashimoto, Hideya; Mashimo, Katsuya. On some 3-dimensional CR submanifolds in $S\sp 6$. Nagoya Math. J. 156 (1999), 171--185.

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