  The index of a Riemannian symmetric space is the minimal codimension of a
proper totally geodesic submanifold (Onishchik, 1980). There is a conjecture by
the first two authors for how to calculate the index. In this paper we give an
affirmative answer to this conjecture for the exceptional Riemannian symmetric
spaces and for the classical symmetric spaces Sp(r,R)/U(r). Our methodology is
new and based on the idea of using slice representations for studying totally
geodesic submanifolds.
