where cn is the volume of the Euclidean unit n-sphere, Yn the nth sectional curvature (see the definition (2) below) and co the volume element of the Riemannian structure of X. It is a still open question, whether the fact that the usual sectional curvature (second order sectional curvature) 72 has a constant sign for all plane sections, has some implications on the sign of 'Yn. Such results would give interesting applications via the generalized Gauss-Bonnet theorem. A known result in this direction is Milnor's theorem (see [2, Theorem 5]), stating that for n = 4 the Euler-Poincare characteristic is positive, if 72 is always positive or always negative. We shall consider the class of Riemannian manifolds arising by division of a compact Lie group G by a closed subgroup H and equipment of the quotient G/H with the invariant Riemannian metric g induced by a bi-invariant metric g on G. Consider the orthogonal decomposition