Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On Curvature-Adapted and Proper Complex Equifocal Sub-manifolds

  • Koike, Naoyuki (Department of Mathematics, Faculty of Science, Tokyo University of Science)
  • Received : 2009.04.13
  • Accepted : 2010.12.09
  • Published : 2010.12.31
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Abstract

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.

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References

  1. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math., 395(1989) 132-141.
  2. J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine Angew. Math. 419(1991), 9-26.
  3. 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. https://doi.org/10.2748/tmj/1113246549
  4. J. Berndt and L. Vanhecke, Curvature adapted submanifolds, Nihonkai Math. J., 3(1992), 177-185.
  5. U. Christ, Homogeneity of equifocal submanifolds, J. Differential Geometry, 62(2002), 1-15. https://doi.org/10.4310/jdg/1090425526
  6. H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math., 35(1934), 588-621. https://doi.org/10.2307/1968753
  7. H. Ewert, A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces, Proc. of Amer. Math. Soc., 126(1998), 2443-2452. https://doi.org/10.1090/S0002-9939-98-04328-7
  8. L. Geatti, Invariant domains in the complexfication of a noncompact Riemannian symmetric space, J. of Algebra, 251(2002), 619-685. https://doi.org/10.1006/jabr.2001.9150
  9. L. Geatti, Complex extensions of semisimple symmetric spaces, manuscripta math., 120(2006), 1-25. https://doi.org/10.1007/s00229-006-0626-1
  10. O. Goertsches and G. Thorbergsson, On the Geometry of the orbits of Hermann actions, Geom. Dedicata, 129(2007), 101-118. https://doi.org/10.1007/s10711-007-9198-9
  11. 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.
  12. 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.
  13. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978.
  14. M. C. Hughes, Complex reflection groups, Communications in Algebra, 18(1990), 3999-4029. https://doi.org/10.1080/00927879008824120
  15. R. Kane, Reflection groups and Invariant Theory, CMS Books in Mathematics, Springer-Verlag, New York, 2001.
  16. N. Koike, Submanifold geometries in a symmetric space of non-compact type and a pseudo-Hilbert space, Kyushu J. Math., 58(2004), 167-202. https://doi.org/10.2206/kyushujm.58.167
  17. N. Koike, Complex equifocal submanifolds and infinite dimensional anti- Kaehlerian isopara-metric submanifolds, Tokyo J. Math., 28(2005), 201-247. https://doi.org/10.3836/tjm/1244208289
  18. N. Koike, Actions of Hermann type and proper complex equifocal submanifolds, Osaka J. Math., 42(2005), 599-611.
  19. N. Koike, A splitting theorem for proper complex equifocal submanifolds, Tohoku Math. J., 58(2006), 393-417. https://doi.org/10.2748/tmj/1163775137
  20. N. Koike, A Chevalley type restriction theorem for a proper complex equifocal submanifold, Kodai Math. J., 30(2007), 280-296. https://doi.org/10.2996/kmj/1183475518
  21. N. Koike, The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry, arXiv:math.DG/0807.1601v2.
  22. A. Kollross, A Classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc., 354(2001), 571-612.
  23. T. Oshima and J. Sekiguchi, The restricted root system of a semisimple symmetric pair, Advanced Studies in Pure Math., 4(1984), 433-497.
  24. R. S. Palais and C.L. Terng, Critical point theory and submanifold geometry, Lecture Notes in Math., 1353, Springer, Berlin, 1988.
  25. W. Rossmann, The structure of semisimple symmetric spaces, Can. J. Math., 1(1979), 157-180.
  26. R. Szoke, Complex structures on tangent bundles of Riemannian manifolds, Math. Ann., 291(1991), 409-428. https://doi.org/10.1007/BF01445217
  27. R. Szoke, Automorphisms of certain Stein manifolds, Math. Z., 219(1995), 357-385. https://doi.org/10.1007/BF02572371
  28. R. Szoke, Adapted complex structures and geometric quantization, Nagoya Math. J., 154(1999), 171-183. https://doi.org/10.1017/S002776300002537X
  29. R. Szoke, Involutive structures on the tangent bundle of symmetric spaces, Math. Ann., 319(2001), 319-348. https://doi.org/10.1007/PL00004437
  30. R. Szoke, Canonical complex structures associated to connections and complexifications of Lie groups, Math. Ann., 329(2004), 553-591. https://doi.org/10.1007/s00208-004-0525-2
  31. C. L. Terng, Isoparametric submanifolds and their Coxeter groups, J. Differential Geometry, 21(1985), 79-107. https://doi.org/10.4310/jdg/1214439466
  32. C. L. Terng and G. Thorbergsson, Submanifold geometry in symmetric spaces, J. Differential Geometry, 42(1995), 665-718. https://doi.org/10.4310/jdg/1214457552

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