Examples of a complex hyperpolar action without singular orbit

• Naoyuki Koike ^[1]
  1. [1] Tokyo University of Science

     Tokyo University of Science

     Japón

     
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 12, Nº. 2, 2010, págs. 127-143
• Idioma: inglés
• DOI: 10.4067/S0719-06462010000200009
• Enlaces
  • Texto completo
• Resumen
  • español

    La noción de una acción hiperpolar compleja sobre un espacio simétrico de tipo no compacto fue recientemente introducida como el análogo de la acción hiperpolar sobre un espacio simétrico de tipo compacto. Como ejemplos de una acción hiperpolar complejas, nosotros tenemos acciones de tipo Hermann, las cuales admiten una orbita (o un punto fijo) singular totalmente geodesica excepto para un ejemplo. Todas las orbitas principales de acciones de tipo Hermann son curvatura-adaptadas y unifocales complejas propias. este artículo, nosotros damos algunos ejemplos de una acción hiperpolar compleja sin orbitas singulares como grupo soluble de acciones libres y encontramos acciones complejas hiperpolares cuyas orbitas son no curvatura-adaptadas o no propias unifocales complejas. También, mostramos que algunos de los ejemplos poseen solamente orbitas minimales.

  • English

    The notion of a complex hyperpolar action on a symmetric space of non-compact type has recently been introduced as a counterpart to the hyperpolar action on a symmetric space of compact type. As examples of a complex hyperpolar action, we have Hermann type actions, which admit a totally geodesic singular orbit (or a fixed point) except for one example. All principal orbits of Hermann type actions are curvature-adapted and proper complex equifocal. In this paper, we give some examples of a complex hyperpolar action without singular orbit as solvable group free actions and find complex hyperpolar actions all of whose orbits are non-curvature-adapted or non-proper complex equifocal among the examples. Also, we show that some of the examples possess the only minimal orbit.

• Referencias bibliográficas
  • Berndt, J. (1998). geneous hypersurfaces in hyperbolic spaces. Math. Z. 229. 589-600
  • Berndt, J,Brück, M. (2001). Cohomogeneity one actions on hyperbolic spaces. J. Reine Angew. Math. 541. 209-235
  • Berndt, J,Tamaru, H. (2003). Homogeneous codimension one foliations on noncompact symmetric space. J. Differential Geometry. 63. 1-40
  • Berndt, J,Tamaru, H. (2004). Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit. Tohoku Math....
  • Berndt, J,Vanhecke, L. (1992). Curvature adapted submanifolds. Nihonkai Math. J. 3. 177-185
  • Christ, U. (2002). Homogeneity of equifocal submanifolds. J. Differential Geometry. 62. 1-15
  • Ewert, H. (1998). A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces. Proc. of Amer. Math. Soc.....
  • Geatti, L. (2002). Invariant domains in the complexfication of a noncompact Riemannian symmetric space. J. of Algebra. 251. 619-685
  • Geatti, L. (2006). Complex extensions of semisimple symmetric spaces. manuscripta math. 120. 1-25
  • Heintze, E,Liu, X,Olmos, C. (2006). Isoparametric submanifolds and a Chevalley type restriction theorem: Integrable systems, geometry, and...
  • Heintze, E,Palais, R.S.,Terng, C.L,Thorbergsson, G,Yau, S. T. (1995). Hyperpolar actions on symmetric spaces, Geometry, topology and physics...
  • Helgason, S. (1978). Differential geometry, Lie groups and symmetric spaces. Academic Press. New York.
  • Koike, N. (2004). Submanifold geometries in a symmetric space of non-compact type and a pseudoHilbert space. Kyushu J. Math. 58. 167-202
  • Koike, N. (2005). Complex equifocal submanifolds and infinite dimensional anti-Kaehlerian isoparametric submanifolds. Tokyo J. Math. 28. 201-247
  • Koike, N. (2005). Actions of Hermann type and proper complex equifocal submanifolds. Osaka J. Math. 42. 599-611
  • Koike, N. (2006). A splitting theorem for proper complex equifocal submanifolds. Tohoku Math. J. 58. 393-417
  • Koike, N. (2007). Complex hyperpolar actions with a totally geodesic orbit. Osaka J. Math. 44. 491-503
  • Kollross, A. (2001). A Classification of hyperpolar and cohomogeneity one actions. Trans. Amer. Math. Soc. 354. 571-612
  • Malcev, A. (1945). On the theory of the Lie groups in the large. Mat. Sb. n. Ser. 16. 163-190
  • Milnor, J. (1976). Curvatures of left invariant metrics on Lie groups. Adv. Math.. 21. 293-329
  • Mostow, G.D. (1961). On maximal subgroups of real Lie groups. Ann. Math. 74. 503-517
  • Palais, R.S,Terng, C.L. (1988). Critical point theory and submanifold geometry. Springer. Berlin.
  • Szöke, R. (1991). Complex structures on tangent bundles of Riemannian manifolds. Math. Ann. 291. 409-428
  • Szöke, R. (1995). Automorphisms of certain Stein manifolds. Math. Z.. 219. 357-385
  • Szöke, R. (1999). Adapted complex structures and geometric quantization. Nagoya Math. J.. 154. 171-183
  • Szöke, R. (2001). Involutive structures on the tangent bundle of symmetric spaces. Math. Ann. 319. 319-348
  • Terng, C.L. (1985). Isoparametric submanifolds and their Coxeter groups. J. Differential Geometry. 21. 79-107
  • Terng, C.L.. (1989). Proper Fredholm submanifolds of Hilbert space. J. Differential Geometry. 29. 9-47
  • Terng, C.L. (1995). Polar actions on Hilbert space. J. Geom. Anal. 5. 129-150
  • Terng, C.L,Thorbergsson, G. (1995). Submanifold geometry in symmetric spaces. J. Differential Geometry. 42. 665-718
  • Wu, B. (1992). Isoparametric submanifolds of hyperbolic spaces. Trans. Amer. Math. Soc. 331. 609-626