Determination of the partial positivity of the curvature in symmetric spaces

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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, ei) > (resp. <) 0, where K(e1, ei) denotes the sectional curvature of the linear span of e1 and ei. 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 Σi=1 (s + 1) K(e1, ei) < (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 languageEnglish
Pages (from-to)107-129
Number of pages23
JournalAnnali di Matematica Pura ed Applicata
Issue number1
StatePublished - 1996


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