We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)le 1. Let g be another metric with scalar curvature k, such that gge g on 2-vectors. We show that kge k everywhere on M implies k=k. Under an additional condition on the Ricci curvature of g, kge k even implies g=g. We also study area-non-increasing spin maps onto such Riemannian manifolds.
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