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# Communications in Analysis and Geometry

## Volume 17 (2009)

### Number 4

### Manifolds with nonnegative isotropic curvature

Pages: 621 – 635

DOI: http://dx.doi.org/10.4310/CAG.2009.v17.n4.a2

#### Author

#### Abstract

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.

Published 1 January 2009