## Abstract

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 {e_{1}, ..., e_{(s + 1)}} of a tangent space at x, Σ _{i = 1}^{(s + 1)} K(e_{1}, e_{i}) > (resp. <) 0, where K(e_{1}, e_{i}) denotes the sectional curvature of the linear span of e_{1} 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(e_{1}, 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.

Original language | English |
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Pages (from-to) | 107-129 |

Number of pages | 23 |

Journal | Annali di Matematica Pura ed Applicata |

Volume | 171 |

Issue number | 1 |

DOIs | |

State | Published - 1996 |