Advanced SearchSearch Tips
On Curvature-Adapted and Proper Complex Equifocal Sub-manifolds
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
  • Journal title : Kyungpook mathematical journal
  • Volume 50, Issue 4,  2010, pp.509-536
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2010.50.4.509
 Title & Authors
On Curvature-Adapted and Proper Complex Equifocal Sub-manifolds
Koike, Naoyuki;
  PDF(new window)
In this paper, we investigate curvature-adapted and proper complex equifocal submanifolds in a symmetric space of non-compact type. The class of these submanifolds contains principal orbits of Hermann type actions as homogeneous examples and is included by that of curvature-adapted and isoparametric submanifolds with flat section. First we introduce the notion of a focal point of non-Euclidean type on the ideal boundary for a submanifold in a Hadamard manifold and give the equivalent condition for a curvature-adapted and complex equifocal submanifold to be proper complex equifocal in terms of this notion. Next we show that the complex Coxeter group associated with a curvature-adapted and proper complex equifocal submanifold is the same type group as one associated with a principal orbit of a Hermann type action and evaluate from above the number of distinct principal curvatures of the submanifold.
proper complex equifocal submanifold;Hermann type action;complex Coxeter group;
 Cited by
The constancy of principal curvatures of curvature-adapted submanifolds in symmetric spaces, Differential Geometry and its Applications, 2014, 35, 103  crossref(new windwow)
J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math., 395(1989) 132-141.

J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine Angew. Math. 419(1991), 9-26.

J. Berndt and H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit, Tohoku Math. J., 56(2004), 163-177. crossref(new window)

J. Berndt and L. Vanhecke, Curvature adapted submanifolds, Nihonkai Math. J., 3(1992), 177-185.

U. Christ, Homogeneity of equifocal submanifolds, J. Differential Geometry, 62(2002), 1-15. crossref(new window)

H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math., 35(1934), 588-621. crossref(new window)

H. Ewert, A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces, Proc. of Amer. Math. Soc., 126(1998), 2443-2452. crossref(new window)

L. Geatti, Invariant domains in the complexfication of a noncompact Riemannian symmetric space, J. of Algebra, 251(2002), 619-685. crossref(new window)

L. Geatti, Complex extensions of semisimple symmetric spaces, manuscripta math., 120(2006), 1-25. crossref(new window)

O. Goertsches and G. Thorbergsson, On the Geometry of the orbits of Hermann actions, Geom. Dedicata, 129(2007), 101-118. crossref(new window)

E. Heintze, X. Liu and C. Olmos, Isoparametric submanifolds and a Chevalley type restriction theorem, Integrable systems, geometry, and topology, 151-190, AMS/IP Stud. Adv. Math. 36, Amer. Math. Soc., Providence, RI, 2006.

E. Heintze, R. S. Palais, C. L. Terng and G. Thorbergsson, Hyperpolar actions on symmetric spaces, Geometry, topology and physics for Raoul Bott (ed. S. T. Yau), Conf. Proc. Lecture Notes Geom. Topology 4, Internat. Press, Cambridge, MA, 1995 pp214-245.

S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978.

M. C. Hughes, Complex reflection groups, Communications in Algebra, 18(1990), 3999-4029. crossref(new window)

R. Kane, Reflection groups and Invariant Theory, CMS Books in Mathematics, Springer-Verlag, New York, 2001.

N. Koike, Submanifold geometries in a symmetric space of non-compact type and a pseudo-Hilbert space, Kyushu J. Math., 58(2004), 167-202. crossref(new window)

N. Koike, Complex equifocal submanifolds and infinite dimensional anti- Kaehlerian isopara-metric submanifolds, Tokyo J. Math., 28(2005), 201-247. crossref(new window)

N. Koike, Actions of Hermann type and proper complex equifocal submanifolds, Osaka J. Math., 42(2005), 599-611.

N. Koike, A splitting theorem for proper complex equifocal submanifolds, Tohoku Math. J., 58(2006), 393-417. crossref(new window)

N. Koike, A Chevalley type restriction theorem for a proper complex equifocal submanifold, Kodai Math. J., 30(2007), 280-296. crossref(new window)

N. Koike, The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry, arXiv:math.DG/0807.1601v2.

A. Kollross, A Classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc., 354(2001), 571-612.

T. Oshima and J. Sekiguchi, The restricted root system of a semisimple symmetric pair, Advanced Studies in Pure Math., 4(1984), 433-497.

R. S. Palais and C.L. Terng, Critical point theory and submanifold geometry, Lecture Notes in Math., 1353, Springer, Berlin, 1988.

W. Rossmann, The structure of semisimple symmetric spaces, Can. J. Math., 1(1979), 157-180.

R. Szoke, Complex structures on tangent bundles of Riemannian manifolds, Math. Ann., 291(1991), 409-428. crossref(new window)

R. Szoke, Automorphisms of certain Stein manifolds, Math. Z., 219(1995), 357-385. crossref(new window)

R. Szoke, Adapted complex structures and geometric quantization, Nagoya Math. J., 154(1999), 171-183. crossref(new window)

R. Szoke, Involutive structures on the tangent bundle of symmetric spaces, Math. Ann., 319(2001), 319-348. crossref(new window)

R. Szoke, Canonical complex structures associated to connections and complexifications of Lie groups, Math. Ann., 329(2004), 553-591. crossref(new window)

C. L. Terng, Isoparametric submanifolds and their Coxeter groups, J. Differential Geometry, 21(1985), 79-107. crossref(new window)

C. L. Terng and G. Thorbergsson, Submanifold geometry in symmetric spaces, J. Differential Geometry, 42(1995), 665-718. crossref(new window)