Manifolds with nonnegative isotropic curvature

Abstract

We prove that if $(M^n,g)$, $n\ge 4$, is a compact, orientable, 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 $\C P^\frac {n}{2}$. (iv) $(M,g)$ is quaternionic-Kähler. This is implied by the following result: Let $(M^{2n},g)$ be a compact, locally irreducible Kähler manifold with nonnegative isotropic curvature. Then either $M$ is biholomorphic to $\C P^n$ or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of S. Brendle and R. Schoen on the Ricci flow.

12 Pages

References in corpus (3)

• Einstein manifolds with nonnegative isotropic curvature are locally symmetric
• On Kähler manifolds with positive orthogonal bisectional curvature
• Ricci flow unstable cell centered at an Einstein metric on the twistor space of positive quaternion Kähler manifolds of dimension $\ge 8$