Nonnegatively and Positively curved manifolds

By Burkhard Wilking. 2007

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The aim of this paper is to survey some results on nonnegatively and positively curved Riemannian manifolds. One of the important features of lower curvature bounds in general is the invariance under taking Gromov Hausdorff limits. Celebrated structure and finiteness results provide a partial understanding of the phenomena that occur while taking limits. These results however are not the subject of this survey since they are treated in other surveys of this volume. In this survey we take the more classical approach and focus on “effective” results. There are relatively few general “effective” structure results in the subject. By Gromov’s Betti number theorem the total Betti number of a nonnegatively curved manifold is bounded above by an explicit constant which only depends on the dimension. The Gromoll Meyer theorem says that a positively curved open manifold is diffeomorphic to the Euclidean space. In the case of nonnegatively curved open manifolds, the soul theorem of Cheeger and Gromoll and Perelman’s solution of the soul conjecture clearly belong to the greatest structure results in the subject, as well. Also relatively good is the understanding of fundamental groups of nonnegatively curved manifolds. A theorem of Synge asserts that an even dimensional orientable compact manifold of positive sectional curvature is simply connected. An odd dimensional positively curved manifold is known to be orientable (Synge), and its fundamental group is finite by the classical theorem of Bonnet and Myers. The fundamental groups of nonnegatively curved manifolds are virtually abelian, as a consequence of Toponogov’s splitting theorem. However, one of the “effective” conjectures in this context, the so called Chern conjecture, was refuted: Shankar [1998] constructed a positively curved manifold with a non cyclic abelian fundamental group. As we will discuss in the last section the known methods for constructing nonnegatively curved manifolds are somewhat limited. The most important tools are the O’Neill formulas which imply that the base of a Riemannian submersion has nonnegative (positive) sectional curvature if the total space has. We recall that a smooth surjective map σ : M → B between two Riemannian manifolds is called a Riemannian submersion if the dual

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/65d8f06cbc9808db31b20e1e312bfc01138fd627

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Citations in the corpus

A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities (2010, methodology): “The existence of this set follows immediately from Böhm and Wilking [2008, proof of Theorem 3.1 combined with Theorem 4.1], see [Theorem 6.1, Wilking 2007] – here we used that C(S, 0) is a Ricci flow invariant curvature condition in between nonnegative curvature operator and nonnegative sectional…”
A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry (2013, unknown): “We recall the following (see also [21]).”