Manifolds with Nonnegative Isotropic Curvature
نویسنده
چکیده
We prove that if (M, g) is a compact locally irreducible Riemannian manifold with nonnegative isotropic curvature, then one of the following possibilities hold: (i) M admits a metric with positive isotropic curvature (ii) (M, g) is isometric to a locally symmetric space (iii) (M, g) is Kähler and biholomorphic to CP n. This is implied by the following two results: (i) Let (M, g) be a compact, locally irreducible Kähler manifold with nonnegative isotropic curvature. Then either M is biholomorphic to CP n or isometric to a compact ireducible Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. (ii) Let (M, g) be a compact, locally irreducible quaternionicKähler manifold with nonnegative isotropic curvature. Then (M, g) is locally symmetric. The proof is based on the recent work of S. Brendle and R. Schoen on the Ricci flow.
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