9 - Manifolds with k-positive Ricci curvature

Summary

Introduction

Let (M, g) be an n-dimensional Riemannian manifold. We say M has k-positive Ricci curvature if at each point p ∈ M the sum of the k smallest eigenvalues of the Ricci curvature at p is positive. We say that the k-positive Ricci curvature is bounded below by α if the sum of the k smallest eigenvalues is greater than α. Note that n-positive Ricci curvature is equivalent to positive scalar curvature and one-positive Ricci curvature is equivalent to positive Ricci curvature. We first describe some basic connect sum and surgery results for k-positive Ricci curvature that are direct generalizations of the well known results for positive scalar curvature (n-positive Ricci curvature). Using these results we construct examples that motivate questions and conjectures in the cases of 2-positive and (n − 1)-positive Ricci curvature. In particular:

Conjecture 1 If M is a closed n-manifold that admits a metric with 2-positive Ricci curvature then the fundamental group, π₁(M), is virtually free.

We formulate an approach to solving this conjecture based, at least philosophically, on the method used in the proof of the Bonnet–Myers theorem: A closed n-manifold that admits a metric with positive Ricci curvature (1-positive Ricci curvature) has finite fundamental group. The Bonnet-Myers theorem proves a diameter bound for a manifold with positive Ricci curvature bounded below and then uses covering spaces to conclude the result on the fundamental group.