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.