The Exceptional Compact Symmetric Spaces G2 and G2/SO(4) as Tubes

摘要

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.