Journal of the Mathematical Society of Japan

The geometry of symmetric triad and orbit spaces of Hermann actions

Osamu IKAWA

Full-text: Open access

Abstract

We introduce the notion of symmetric triad, which is a generalization of the notion of irreducible root system, and study its fundamental properties. Applying these results, we study the orbit spaces of Hermann actions on compact symmetric spaces.

Article information

Source
J. Math. Soc. Japan, Volume 63, Number 1 (2011), 79-136.

Dates
First available in Project Euclid: 27 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1296138345

Digital Object Identifier
doi:10.2969/jmsj/06310079

Mathematical Reviews number (MathSciNet)
MR2752433

Zentralblatt MATH identifier
1213.53064

Subjects
Primary: 53C35: Symmetric spaces [See also 32M15, 57T15]
Secondary: 17B22: Root systems 53C40: Global submanifolds [See also 53B25]

Keywords
Hermann action austere submanifold symmetric triad

Citation

IKAWA, Osamu. The geometry of symmetric triad and orbit spaces of Hermann actions. J. Math. Soc. Japan 63 (2011), no. 1, 79--136. doi:10.2969/jmsj/06310079. https://projecteuclid.org/euclid.jmsj/1296138345


Export citation

References

  • N. Bourbaki, Groupes et algebres de Lie, Hermann, Paris, 1975.
  • R. L. Bryant, Some remarks on the geometry of austere manifolds, Bol. Soc. Bras. Mat., 21 (1991), 133–157.
  • O. Goertsches and G. Thorbergsson, On the geometry of orbits of Hermann actions, Geom. Dedicata, 129 (2007), 101–118.
  • E. Heintze, R. S. Palais, C. Terng and G. Thobergsson, Hyperpolar actions on symmetric spaces, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp.,214–245.
  • R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math., 148 (1982), 47–157.
  • S. Helgason, Differential Geometry, Lie groups, and symmetric spaces, Academic Press, 1978.
  • D. Hirohashi, O. Ikawa and H. Tasaki, Orbits of isotropy groups of compact symmetric spaces, Tokyo J. Math., 24 (2001), 407–428.
  • D. Hirohashi, H. Tasaki, H. Song and R. Takagi, Minimal orbits of the isotropy groups of symmetric spaces of compact type, Differential Geom. Appl., 13 (2000), 167–177.
  • O. Ikawa, Equivariant minimal immersions of compact Riemannian homogeneous spaces into compact Riemannian homogeneous spaces, Tsukuba J. Math., 17 (1993), 169–188.
  • O. Ikawa, T. Sakai and H. Tasaki, Orbits of Hermann actions, Osaka J. Math., 38 (2001), 923–930.
  • O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds and austere submanifolds, J. Math. Soc. Japan, 61 (2009), 437–481.
  • S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, II, Reprint of the 1969 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996.
  • N. Koike, The mean curvature flow for equifocal submanifolds, arXiv:math. DG/0901.2432v2
  • A. Kollross, A classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc., 354 (2002), 571–612.
  • Dominic S. P. Leung, The reflection principle for minimal submanifolds of Riemannian symmetric spaces, J. Differential Geometry, 8 (1973), 153–160.
  • T. Matsuki, Double coset decompositions of reductive Lie groups arising from two involutions, J. Algebra, 197 (1997), 49–91.
  • T. Matsuki, Classification of two involutions on compact semisimple Lie groups and root systems, J. Lie Theory, 12 (2002), 41–68.
  • M. Takeuchi, Modern spherical functions, Translations of mathematical monographs, 135, Amer. Math. Soc.
  • J. A. Wolf, Complex homogeneous contact manifolds and quaterninonic symmetric spaces, J. Math. Mech., 14 (1965), 1033–1047.