Manifolds with nonnegative isotropic curvature

We prove that if $(M^n,g)$, $n \ge 4$, is a compact, orientable,locally irreducible Riemannian manifold with nonnegative isotropiccurvature, then one of the following possibilities hold:\leftskip-3pt\begin{enumerate}\item[(i)] $M$ admits a metric with positive isotropic curvature.\item[(ii)] $(M,g)$ is isometric to a locally symmetric space.\item[(iii)] $(M,g)$ is Kähler and biholomorphic to ${\mathbb C} P^\frac{n}{2}$.\item[(iv)] $(M,g)$ is quaternionic-Kähler.\end{enumerate}

This is implied by the following result:

Let $(M^{2n},g)$ be a compact, locally irreducible Kählermanifold with nonnegative isotropic curvature. Then either $M$ isbiholomorphic to ${\mathbb C} P^n$ or isometric to a compactHermitian symmetric space. This answers a question of Micallef andWang in the affirmative.

The proof is based on the recent work of Brendle and Schoen on theRicci flow.

Full Text (PDF format)

Published 1 January 2009