Manifolds with nonnegative isotropic curvature

Details Manifolds with nonnegative isotropic curvature Manifolds with nonnegative isotropic curvature

2007-07-26

arxiv.org/abs/0707.3894v4

Communications in Analysis and Geometry 17 (2009), no. 4, 621-635

Mathematics

Differential Geometry

12 Pages

Scientific paper

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 K\"ahler and biholomorphic to $\C P^\frac {n}{2}$. (iv) $(M,g)$ is quaternionic-K\"ahler. This is implied by the following result: Let $(M^{2n},g)$ be a compact, locally irreducible K\"ahler 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.

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