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Polar Actions on Symmetric Spaces

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Recent Trends in Lorentzian Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 26))

Abstract

We present the classification results for polar actions on symmetric spaces known to date.

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References

  1. Adams, S., Stuck, G.: The isometry group of a compact Lorentz manifold I. Invent. Math. 129, 239–261 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bergmann, I.: Reducible polar representations. Manuscripta Math. 104, 309–324 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berndt, J., Console, S., Olmos, C.: Submanifolds and holonomy. Chapman & Hall/CRC Research Notes in Mathematics, vol. 434, Chapman & Hall/CRC, Boca Raton, FL (2003)

    Google Scholar 

  4. Berndt, J., Díaz-Ramos, J.C.: Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. Proc. Amer. Math. Soc. 135, 3349–3357 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berndt, J., Díaz-Ramos, J.C.: Homogeneous hypersurfaces in complex hyperbolic spaces. Geom. Dedicata 138, 129–150 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berndt, J., Díaz-Ramos, J. C.: Homogeneous polar foliations of complex hyperbolic spaces. Comm. Anal. Geom. 20(3), 435–454 (2012)

    Google Scholar 

  7. Berndt, J., Díaz-Ramos, J.C.: Polar actions on the complex hyperbolic plane. Ann. Glob. Anal. Geom. DOI 10.1007/s10455-012-9335-z

    Google Scholar 

  8. Berndt, J., Díaz-Ramos, J.C., Tamaru, H.: Hyperpolar homogeneous foliations on symmetric spaces of noncompact type. J. Differential Geom. 86, 191–235 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Berndt, J., Tamaru, H.: Cohomogeneity one actions on noncompact symmetric spaces of rank one. Trans. Amer. Math. Soc. 359, 3425–3438 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Berndt, J., Tamaru, H.: Cohomogeneity one actions on symmetric spaces of noncompact type. To appear in J. Reine Angew. Math. Preprint arXiv:1006.1980 [math.DG].

    Google Scholar 

  11. Biliotti, L.: Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces. Trans. Amer. Math. Soc. 358, 3003–3022 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Biliotti, L., Gori, A.: Coisotropic and polar actions on complex Grassmannians. Trans. Amer. Math. Soc. 357, 1731–1751 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bott, R., Samelson, H.: Applications of the theory of Morse to symmetric spaces. Amer. J. Math. 80, 964–1029 (1958)

    Article  MathSciNet  Google Scholar 

  14. Conlon, L.: Variational completeness and K-transversal domains. J. Differ. Geom. 5, 135–147 (1971)

    MathSciNet  MATH  Google Scholar 

  15. Dadok, J.: Polar coordinates induced by actions of compact Lie groups. Trans. Amer. Math. Soc. 288, 125–137 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dadok, J., Kac, V.: Polar representations. J. Algebra 92, 504–524 (1985)

    MathSciNet  MATH  Google Scholar 

  17. D’Ambra, G.: Isometry groups of Lorentz manifolds. Invent. Math. 92, 555–565 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. D’Ambra, G., Gromov, M.: Lectures on transformation groups: geometry and dynamics. Surveys in differential geometry, Cambridge, MA, (1990), pp. 19–111, Lehigh University Bethlehem, PA (1991)

    Google Scholar 

  19. Díaz-Ramos, J.C.: Proper isometric actions. arXiv:0811.0547v1 [math.DG]

    Google Scholar 

  20. Díaz-Ramos, J.C., Domínguez-Vázquez, M.: Isoparametric hypersurfaces in Damek-Ricci spaces. arXiv:1111.0264v1 [math.DG]

    Google Scholar 

  21. Díaz-Ramos, J.C., Kollross, A.: Polar actions with a fixed point. Differ. Geom. Appl. 29, 20–25 (2011)

    Article  MATH  Google Scholar 

  22. Eschenburg, J.-H., Heintze, E.: Polar representations and symmetric spaces. J. Reine Angew. Math. 507, 93–106 (1999)

    MathSciNet  MATH  Google Scholar 

  23. Eschenburg, J.-H., Heintze, E.: On the classification of polar representations. Math. Z. 232, 391–398 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Geatti, L., Gorodski, C.: Polar orthogonal representations of real reductive algebraic groups. J. Algebra 320, 3036–3061 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gorodski, C., Thorbergsson, G.: Variationally complete actions on compact symmetric spaces. J. Differ. Geom. 62, 39–48 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Heintze, E., Palais, R.S., Terng, C.-L., Thorbergsson, G.: Hyperpolar actions on symmetric spaces. Geometry, topology, and physics, Conf. Proc. Lecture Notes Geom. Topology, IV, pp. 214–245. Int. Press, Cambridge (1995)

    Google Scholar 

  27. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original, Graduate Studies in Mathematics, vol. 34. American Mathematical Society, Providence (2001)

    Google Scholar 

  28. Hermann, R.: Variational completeness for compact symmetric spaces, Proc. Amer. Math. Soc. 11, 544–546 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hsiang, W.-Y., Lawson Jr., H.B.: Minimal submanifolds of low cohomogeneity. J. Differ. Geom. 5, 1–38 (1971)

    MathSciNet  MATH  Google Scholar 

  30. Iwata, K.: Classification of compact transformation groups on cohomology quaternion projective spaces with codimension one orbits. Osaka J. Math. 15, 475–508 (1978)

    MathSciNet  MATH  Google Scholar 

  31. Iwata, K.: Compact transformation groups on rational cohomology Cayley projective planes. Tôhoku Math. J. 33(2), 429–442 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  32. Knapp, A.W.: Lie Groups Beyond an Introduction. Birkhäuser, Boston (1996)

    MATH  Google Scholar 

  33. Kollross, A.: A classification of hyperpolar and cohomogeneity one actions. Trans. Amer. Math. Soc. 354, 571–612 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kollross, A.: Polar actions on symmetric spaces. J. Differ. Geom. 77, 425–482 (2007)

    MathSciNet  MATH  Google Scholar 

  35. Kollross, A.: Low cohomogeneity and polar actions on exceptional compact Lie groups. Transform. Groups 14, 387–415 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kollross, A.: Duality of symmetric spaces and polar actions. J. Lie Theory 21, 961–986 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Kollross, A., Lytchak, A.: Polar actions on symmetric spaces of higher rank. arXiv:1108. 3256v2 [math.DG]

    Google Scholar 

  38. Loos, O.: Symmetric spaces. I: General Theory. II: Compact spaces and classification. W.A. Benjamin (1969)

    Google Scholar 

  39. Lytchak, A., Thorbergsson, G.: Variationally complete actions on nonnegatively curved manifolds. Illinois J. Math. 51, 605–615 (2007)

    MathSciNet  MATH  Google Scholar 

  40. Montgomery, D., Yang, C.T.: The existence of a slice. Ann. Math. 65(2), 108–116 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  41. Myers, S.B., Steenrod, N.E.: The group of isometries of a Riemannian manifold. Ann. Math. 40(2), 400–416 (1939)

    Article  MathSciNet  Google Scholar 

  42. Palais, R.S., On the existence of slices for actions of non-compact Lie groups. Ann. Math. 73(2), 295–323 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  43. Palais, R.S., Terng, C.-L.: A general theory of canonical forms. Trans. Amer. Math. Soc. 300, 771–789 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  44. Podestà, F., Thorbergsson, G.: Polar actions on rank-one symmetric spaces. J. Differ. Geom. 53, 131–175 (1999)

    MATH  Google Scholar 

  45. Podestà, F., Thorbergsson, G.: Polar and coisotropic actions on Kähler manifolds. Trans. Amer. Math. Soc. 354, 1759–1781 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Takagi, R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10, 495–506 (1973)

    MathSciNet  MATH  Google Scholar 

  47. Takagi, R., Takahashi, T.: On the principal curvatures of homogeneous hypersurfaces in a sphere. Differential Geometry (in Honor of Kentaro Yano), pp. 469–481. Kinokuniya, Tokyo (1972)

    Google Scholar 

  48. Terng, C.-L.: Polar actions on Hilbert space, J. Geom. Anal. 5, 129–150 (1995)

    Article  MathSciNet  Google Scholar 

  49. Thorbergsson, G.: A survey on isoparametric hypersurfaces and their generalizations. Handbook of Differential Geometry, vol. I, pp. 963–995. North-Holland, Amsterdam (2000)

    Google Scholar 

  50. Thorbergsson, G.: Transformation groups and submanifold geometry. Rendiconti di Matematica, Serie VII, 25, 1–16 (2005)

    MathSciNet  MATH  Google Scholar 

  51. Thorbergsson, G.: Singular Riemannian foliations and isoparametric submanifolds. Milan J. Math. 78, 355–370 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wu, B.: Isoparametric submanifolds of hyperbolic spaces. Trans. Amer. Math. Soc. 331, 609–626 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  53. Yau, S.T.: Remarks on the group of isometries of a Riemannian manifold. Topology 16, 239–247 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Santiago de Compostela, Santiago de Compostela, Spain

    José Carlos Díaz-Ramos

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  1. José Carlos Díaz-Ramos
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Correspondence to José Carlos Díaz-Ramos .

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Editors and Affiliations

  1. , Dept of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E18071, Granada, Spain

    Miguel Sánchez

  2. Departamento de Mátematica Aplicada, Universidad de Granada, Campus de Fuentenueva, Granada, 18071, Spain

    MIguel Ortega

  3. , Dept. of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E-18071, Granada, Spain

    Alfonso Romero

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Díaz-Ramos, J.C. (2012). Polar Actions on Symmetric Spaces. In: Sánchez, M., Ortega, M., Romero, A. (eds) Recent Trends in Lorentzian Geometry. Springer Proceedings in Mathematics & Statistics, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4897-6_14

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