Ricci-Positive Metrics on Connected Sums of Projective Spaces

Abstract

It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed positive Ricci metrics on \(\#_k\mathbf{C}P^2\). In this paper, we revisit and extend Perelman's construction to show that \(\#_k\mathbf{C}P^n\), \(\#_k\mathbf{H}P^n\), and \(\#_k\mathbf{O}P^2\) all admit metrics of positive Ricci curvature.