Abstract.
In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G 2 and G 2/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G 2 and G 2/SO(4) can be thought of as compact tubes around SU(3) and ℂP 2, respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.
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The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.
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Verhóczki, L. The Exceptional Compact Symmetric Spaces G 2 and G 2/SO(4) as Tubes. Monatsh. Math. 141, 323–335 (2004). https://doi.org/10.1007/s00605-002-0036-8
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DOI: https://doi.org/10.1007/s00605-002-0036-8