On compact symmetric spaces associated to the exceptional Lie group $E_6$

Abstract

We discuss cohomogeneity one isometric actions on the exceptional compact symmetric spaces$E_6/(SU(6)\times SU(2)/\mathbb{Z}_2),\E_6/(Spin(10)\times U(1)/\mathbb{Z}_4)$ and $E_6/F_4$.These symmetric spaces can be thought of as compact tubes, since the principal orbits of the considered isometricactions coincide with tubular hypersurfaces aroundtotally geodesic singular orbits.We determine the radii of the tubes andthe principal curvatures of the tubular hypersurfaces.Moreover, we compute the volumesof the principal orbits and the volumes of the symmetric spacesin terms of the maximal sectional curvature.