Manuscripta Mathematica

, Volume 141, Issue 1–2, pp 51–62 | Cite as

Symmetric spaces as Grassmannians

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Abstract

In the PhD thesis of Huang with Leung (Huang, A uniform description of Riemannian symmetric spaces as Grassmannians using magic square, www.ims.cuhk.edu.hk/~leung/; Huang and Leung, Math Ann 350:76–106, 2010), all compact symmetric spaces are represented as (structured) Grassmannians over the algebra \({\mathbb{K L}:=\mathbb{K} \otimes_{\mathbb{R}} \mathbb{L}}\) where \({\mathbb{K},\mathbb{L}}\) are real division algebras. This was known in some (infinitesimal) sense for exceptional spaces (see Baez, Bull Am Math Soc 39:145–205, 2001); the main purpose in Huang (www.ims.cuhk.edu.hk/~leung/) and Huang and Leung (Math Ann 350:76–106, 2010) was to give a similar description for the classical spaces. In the present paper we give a different approach to this result by investigating the fixed algebras \({\mathbb{B}}\) of involutions on \({\mathbb{A} =\mathbb{K}\mathbb{L}}\) with half-dimensional eigenspaces together with the automorphism groups of \({\mathbb{A}}\) and \({\mathbb{B}}\). We also relate the results to the classification of self-reflective submanifolds in Chen and Nagano (Trans Am Math Soc 308:273–297, 1988) and Leung (J Differ Geom 14:167–177, 1979).

Mathematics Subject Classification (2000)

53C35 53C40 17A35 14M15 

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References

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    Baez, J.: The octonians. Bull. Am. Math. Soc 39, 145–205 (2001). http://mathhttp://math.ucr.edu/home/baez/octonions
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    Chen B.-Y., Nagano T.: A Riemannian geometric invariant and its applications to a problem of Borel and Serre. Trans. Am. Math. Soc 308, 273–297 (1988)MathSciNetMATHCrossRefGoogle Scholar
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    Helgason S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York (1978)MATHGoogle Scholar
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    Huang Y.: A uniform description of Riemannian symmetric spaces as Grassmannians using magic square. http://www.ims.cuhk.edu.hk/~leung/
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    Huang Y., Leung N.C.: A uniform description of compact symmetric spaces as Grassmannians using magic square. Math. Ann. 350, 76–106 (2010)MathSciNetGoogle Scholar
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    Leung D.S.P.: Reflective submanifolds III. J. Differ. Geom. 14, 167–177 (1979)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institut für MathematikUniversität AugsburgAugsburgGermany
  2. 2.Department of MathematicsUniversity of IsfahanIsfahanIran

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