Determination of the partial positivity of the curvature in symmetric spaces

This paper is devoted to the study of the partial positivity (resp. negativity) of the curvature of irreducible symmetric spaces of the classical types. A Riemannian manifold M is called to have s-positive (resp. s-negative) curvature if for each x ε M and for any (s + 1) or-thonormal vectors {e₁, ..., e_{(s + 1)}} of a tangent space at x, Σ _{i = 1}^{(s + 1)} K(e₁, e_i) > (resp. <) 0, where K(e₁, e_i) denotes the sectional curvature of the linear span of e₁ and e_i. If M is an irreducible symmetric space of the compact (resp. noncompact) type, then there exists the small-est integer s, 1 ≤ s < n such that Σ_i=1 ^{(s + 1)} K(e₁, e_i) < (resp. <) 0, since M has nonnegative (resp. nonpositive) sectional curvature and positive (resp. negative) Ricci curvature. Here, we compute this integer s for all the irreducible classical symmetric spaces of the compact (resp. noncompact) type. In general, knowing this s leads to important geometric as well as purely topological consequences.