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Symmetric submanifolds associated with irreducible symmetric R-spaces

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Abstract.

We construct examples of symmetric submanifolds in Riemannian symmetric spaces of noncompact type and obtain the classification of symmetric submanifolds in irreducible Riemannian symmetric spaces of noncompact type and rank greater than one. This finishes the classification problem of symmetric submanifolds in Riemannian symmetric spaces completely.

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References

  1. Eschenburg, J.H.: Higher rank curved Lie triples. J. Math. Soc. Japan 54, 551–564 (2002)

    Google Scholar 

  2. Ferus, D.: Symmetric submanifolds of Euclidean space. Math. Ann. 247, 81–93 (1980)

    Google Scholar 

  3. Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Google Scholar 

  4. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Academic Press, San Diego, 1978

  5. Kobayashi, S., Nagano, T.: On filtered Lie algebras and geometric structures I. J. Math. Mech. 13, 875–907 (1964)

    Google Scholar 

  6. Kobayashi, S., Nomizu, K.: Foundations of differential geometry II, Wiley, New York, 1969

  7. Leung, D.S.P.: On the classification of reflective submanifolds of Riemannian symmetric spaces, Indiana Univ. Math. J. 24, 327–339 (1974); Errata: ibid. 24, 1199 (1975)

    Google Scholar 

  8. Leung, D.S.P.: Reflective submanifolds. III. Congruency of isometric reflective submanifolds and corrigenda to the classification of reflective submanifolds. J. Differential Geom. 14, 167–177 (1979)

    CAS  Google Scholar 

  9. Nagano, T.: Transformation groups on compact symmetric spaces. Trans. Am. Math. Soc. 118, 428–453 (1965)

    Google Scholar 

  10. Naitoh, H.: Symmetric submanifolds of compact symmetric spaces. Tsukuba J. Math. 10, 215–242 (1986)

    Google Scholar 

  11. Naitoh, H.: Compact simple Lie algebras with two involutions and submanifolds of compact symmetric spaces I, II. Osaka J. Math. 30, 653–690, 691–732 (1993)

    Google Scholar 

  12. Naitoh, H.: Grassmann geometries on compact symmetric spaces of general type. J. Math. Soc. Japan 50, 557–592 (1998)

    Google Scholar 

  13. Naitoh, H.: Grassmann geometries on compact symmetric spaces of exceptional type. Japanese J. Math. 26, 157–206 (2000)

    Google Scholar 

  14. Naitoh, H.: Grassmann geometries on compact symmetric spaces of classical type. Japanese J. Math. 26, 219–319 (2000)

    Google Scholar 

  15. Naitoh, H., Takeuchi, M.: Symmetric submanifolds of symmetric spaces. Sugaku 2, 157–188 (1989)

    Google Scholar 

  16. Osipova, D.: Symmetric submanifolds in symmetric spaces. Diff. Geom. Appl. 16, 199–211 (2002)

    Google Scholar 

  17. Takeuchi, M.: Cell decompositions and Morse equalities on certain symmetric spaces. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 12, 81–192 (1965)

    Google Scholar 

  18. Takeuchi, M.: Parallel submanifolds of space forms, in: Manifolds and Lie groups, Pap. in Honor of Y. Matsushima, Prog. Math. 14, 429–447 (1981)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University College, Cork, Ireland

    Jürgen Berndt

  2. Institut für Mathematik, Universität Augsburg, 86135, Augsburg, Germany

    Jost-Hinrich Eschenburg

  3. Department of Mathematics, Yamaguchi University, 753-8512, Yamaguchi, Japan

    Hiroo Naitoh

  4. Department of Mathematics, Ochanomizu University, 112-8610, Tokyo, Japan

    Kazumi Tsukada

Authors
  1. Jürgen Berndt
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  2. Jost-Hinrich Eschenburg
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  3. Hiroo Naitoh
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  4. Kazumi Tsukada
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Correspondence to Jürgen Berndt.

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Berndt, J., Eschenburg, JH., Naitoh, H. et al. Symmetric submanifolds associated with irreducible symmetric R-spaces. Math. Ann. 332, 721–737 (2005). https://doi.org/10.1007/s00208-005-0646-2

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  • DOI: https://doi.org/10.1007/s00208-005-0646-2

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