Classification of manifolds with weakly 1/4-pinched curvatures

By S. Brendle, R. Schoen. 2007

166 citations recorded by the source corpus.

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R2 has non-negative isotropic curvature.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a4033d8a581939eb8ead44587856171b001f27d0

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

An extension of Mok’s theorem on the generalized Frankel conjecture (2007, methodology): “Recently, based on the work of Brendle-Schoen [2], the first author [5] gave a simple and completely transcendental proof to Mok’s theorem on the generalized Frankel conjecture.”
Complete classification of compact four-manifolds with positive isotropic curvature (2008, methodology): “By the remark after the statement of Hamilton’s Theorem, the conclusion in the conjecture of Schoen holds.”
The sphere theorems for manifolds with positive scalar curvature (2011, methodology): “[8] S. Brendle and R. Schoen, Sphere theorems in geometry, Surveys in Differential Geometry, Vol.13, 2009, 49-84.”
Manifolds with 1/4-pinched curvature are space forms (2007, methodology): “The techniques in this paper can be extended to give a classification of manifolds with weakly 1/4-pinched sectional curvatures. We refer to [ 4 ] for details.”
Topological and differentiable sphere theorems for complete submanifolds (2009, methodology): “Recently, Brendle and Schoen [10, 11] proved a remarkable differentiable pinching theorem for point-wise 1/4-pinched Riemannian manifolds by developing the theory and techniques of Ricci flow [22–24].”
A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities (2010, methodology): “We can now apply Proposition 4 from [Brendle and Schoen, 2008], to see that the level set ũ(0) is invariant under the (local) flows of the vectorfields Xi.”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, background): “[5, 23, 30]), it is natural to ask whether there is a parabolic flow of metrics on M which starts at g0, preserves the Hermitian condition, and reveals information about the structure ofM as a complex manifold.”
How to produce a Ricci Flow via Cheeger-Gromoll exhaustion (2011, background): “Thus, combining with the knowledge from [7] of the compact case, we extend the same classification of [7] to open manifolds withKC ≥ 0; more precisely, any such manifold admits a complete non-negatively curved locally symmetric metric ĝ, i.”
Frankel conjecture and Sasaki geometry (2012, methodology): “In a sequel [37], we will use the results of this paper together with the technique of Brendle-Schoen [10] (c.”
Geometric, topological and differentiable rigidity of submanifolds in space forms (2011, background): “[8] S. Brendle and R. Schoen, Sphere theorems in geometry, Surveys in Differential Geometry, Vol.13, 2009, 49-84.”