Contents Online
Communications in Analysis and Geometry
Volume 17 (2009)
Number 4
Manifolds with nonnegative isotropic curvature
Pages: 621 – 635
DOI: https://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