Mathematics > Differential Geometry
[Submitted on 26 Jul 2007 (v1), last revised 8 Apr 2011 (this version, v4)]
Title:Manifolds with nonnegative isotropic curvature
View PDFAbstract: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.
Submission history
From: Harish Seshadri [view email][v1] Thu, 26 Jul 2007 10:43:21 UTC (6 KB)
[v2] Mon, 17 Sep 2007 04:15:21 UTC (10 KB)
[v3] Wed, 14 May 2008 17:29:35 UTC (15 KB)
[v4] Fri, 8 Apr 2011 10:50:49 UTC (13 KB)
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