Manifolds with Pointwise 1/4-pinched Curvature
نویسنده
چکیده
In this lecture we will describe our recent joint work with SimonBrendle ([1], [2]) in which we give the differentiable classification ofcompact Riemannian manifolds with pointwise 1/4-pinched curvature.Our theorems are:Theorem 1. Let M be a compact Riemannian manifold with pointwise1/4-pinched curvature. Then M admits a metric of constant curvature,and therefore is diffeomorphic to a spherical space form.Theorem 2. Let M be a compact Riemannian manifold with weaklypointwise 1/4-pinched sectional curvatures. It then follows that either:i) M is isometric to a rank 1 locally symmetric space, or ii) M isdiffeomorphic to a spherical space form.References[1] S. Brendle and R. Schoen, Manifolds with 1/4-pinched curvature are spaceforms, preprint (2007)[2] S. Brendle and R. Schoen, Classification of manifolds with weakly 1/4-pinchedcurvatures, preprint (2007) Department of Mathematics, Stanford University, Stanford, CA94305
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