Volumes of Compact Symmetric Spaces

H. Freudenthal [4] defined the natural volume of a semi-simple compact Lie group $G$ induced from the Killing form and gave a formula of the natural volume of $G$ . S. A. Broughton [3] calculated the volume in the case of $G$ a classical Lie group. Another volume formula of the semi-simple compact Lie groups has been studied by H. Urakawa [7] and I. G. Macdonald [6] in different ways, respectively. If $G/K$ is a compact symmetric space, then the Killing form of $G$ also induces a natural volume of $G/K$. The volumes of the projective spaces are obtained by using Jacobi fields (cf. [2], [5]). In the previous paper [1] we calculated the volumes of the Hermitian exceptional symmetric spaces EIII, EVII and the twister space $Z(EIX)$ of the exceptional symmetric space $EIX$ by using the computations of the 1st Chem classes. From those results we can calculate the natural volumes of the compact symmetric spaces as follows: symbol space