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On Totally Geodesic Surfaces in Symmetric Spaces of Type AI

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Real and Complex Submanifolds

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 106))

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Abstract

We develop an approach to the classification of nonflat totally geodesic surfaces in Riemannian symmetric spaces of noncompact type. In this paper, we concentrate on the case of symmetric spaces of type AI, and show that such surfaces correspond to certain nilpotent matrices. As applications, we obtain explicit classifications in the cases of rank two and three.

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Acknowledgements

The authors would like to thank Yoshio Agaoka, Katsuya Mashimo, and Takayuki Okuda for useful comments and discussions. The third author was supported in part by KAKENHI (24654012).

Author information

Authors and Affiliations

  1. Department of Mathematics, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

    Takuya Fujimaru, Akira Kubo & Hiroshi Tamaru

  2. Usuki High School, Usuki, 875-0042, Oita, Japan

    Takuya Fujimaru

Authors
  1. Takuya Fujimaru
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  2. Akira Kubo
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  3. Hiroshi Tamaru
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Corresponding author

Correspondence to Hiroshi Tamaru .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Kyungpook National University, Daegu, Republic of Korea

    Young Jin Suh

  2. Department of Mathematics, King’s College London, London, UK

    Jürgen Berndt

  3. Department of Mathematics, Osaka City University, Osaka, Japan

    Yoshihiro Ohnita

  4. Department of Applied Mathematics, Kyung Hee University, Yongin, Republic of Korea

    Byung Hak Kim

  5. The Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang, Republic of Korea

    Hyunjin Lee

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Fujimaru, T., Kubo, A., Tamaru, H. (2014). On Totally Geodesic Surfaces in Symmetric Spaces of Type AI. In: Suh, Y.J., Berndt, J., Ohnita, Y., Kim, B.H., Lee, H. (eds) Real and Complex Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 106. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55215-4_19

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