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Geometry of G2 orbits and isoparametric hypersurfaces

Published online by Cambridge University Press:  11 January 2016

Reiko Miyaoka
Reiko Miyaoka*
Affiliation:
Mathematical Institute, Graduate School of Sciences, Tohoku University, Aoba-ku, Sendai [email protected]
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Abstract

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We characterize the adjoint G2 orbits in the Lie algebra g of G2 as fibered spaces over S6 with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case (g,m) = (6,2) with case (3,2). The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations of S6 and G2/SO(4). From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.

Keywords

53C4053C3032L25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 203 , September 2011 , pp. 175 - 189
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

References

[A] Abresch, U., Isoparametric hypersurfaces with four and six distinct principal curvatures, Math. Ann. 264 (1983), 283–302.CrossRefGoogle Scholar
[B1] Bryant, R., Submanifolds and special structures on the octonions, J. Differential Geom. 17 (1982), 185–232.CrossRefGoogle Scholar
[B2] Bryant, R., Lie groups and twistor spaces, Duke Math. J. 52 (1985), 223–261.CrossRefGoogle Scholar
[C] Cartan, E., Familles de surfaces isoparamétriques dan les espaces à courbure constante, Ann. of Math. (2) 17 (1938), 177–191.Google Scholar
[CCJ] Cecil, T. E., Chi, Q.-S., and Jensen, G. R., Isoparametric hypersurfaces with four principal curvatures, Ann. of Math. (2) 166 (2007), 1–76.CrossRefGoogle Scholar
[DN] Dorfmeister, J. and Neher, E., Isoparametric hypersurfaces, case g = 6, m = 1, Comm. Algebra 13 (1985), 2299–2368.CrossRefGoogle Scholar
[FKM] Ferus, D., Karcher, H., and Münzner, H. F., Cliffordalgebren und neue isoparametrische Hyperfl¨achen, Math. Z. 177 (1981), 479–502.CrossRefGoogle Scholar
[HL] Hsiang, W. Y. and Lawson, H. B., Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5 (1971), 1–38.CrossRefGoogle Scholar
[I] Immervoll, S., The classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres, Ann. of Math. (2) 168 (2008), 1011–1024.CrossRefGoogle Scholar
[MO] Ma, H. and Ohnita, Y., Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces, preprint, 2010.Google Scholar
[M1] Miyaoka, R., The linear isotropy group of G2/SO(4), the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. 30 (1993), 179–202.Google Scholar
[M2] Miyaoka, R., Isoparametric geometry and related fields, Adv. Stud. Pure Math. 51 (2008), 305–326.Google Scholar
[M3] Miyaoka, R., The Dorfmeister-Neher theorem on isoparametric hypersurfaces, Osaka J. Math. 46 (2009), 1–21.Google Scholar
[M4] Miyaoka, R., Isoparametric hypersurfaces with (g,m) = (6, 2), preprint, 2009.Google Scholar
[Muü] Muünzner, H. F., Isoparametrische Hyperfläachen in Sphäaren I, Math. Ann. 251 (1980), 57–71; II, 256 (1981), 215–232.CrossRefGoogle Scholar
[OT] Ozeki, H. and Takeuchi, M., On some types of isoparametric hypersurfaces in spheres, II, Tohoku Math. J. (2) 28 (1976), 7–55.CrossRefGoogle Scholar
[TT] Takagi, R. and Takahashi, T., “On the principal curvatures of homogeneous hypersurfaces in a sphere” in Differential Geometry (in Honor of K. Yano), Kinokuniya, Tokyo, 1972, 469–481.Google Scholar
[Th] Thorbergsson, G., “A survey on isoparametric hypersurfaces and their generalizations” in Handbook of Differential Geometry, I, North-Holland, Amsterdam, 2000, 963–995.CrossRefGoogle Scholar
[Y] Yau, S.-T., “Open problems in geometry” in S. S. Chern—A Great Geometer of the Twentieth Century, International Press, Hong Kong, 1992, 275–319.Google Scholar