Tubular Structures of Compact Symmetric Spaces Associated with the Exceptional Lie Group F4

In the present paper we discuss in detail cohomogeneity one isometric actions on the compact symmetric spaces F4 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{4}\backslash({\rm Sp}(3)\,\times\, {\rm Sp}(1)\backslash {\mathbb Z} _2)$$\end{document} . By means of isometric actions we show that these symmetric spaces have got well-defined tubular structures around the totally geodesic submanifolds Spin(9) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G^+_4({\mathbb R}^9)$$\end{document} , respectively. Therefore these symmetric spaces can be thought of as compact tubes. The radii of the tubes and the principal curvatures of the tubular hypersurfaces are determined in explicit form. Moreover, we apply these results to compute the volumes of the principal orbits and the volumes of the symmetric spaces associated with the compact Lie group F4.