# Ricci-Positive Metrics on Connected Sums of Projective Spaces

## TL;DR

This paper extends Perelman's techniques to construct positive Ricci curvature metrics on connected sums of various projective spaces, showing such manifolds can have unbounded Betti numbers unlike those with positive sectional curvature.

## Contribution

It introduces new methods to build positive Ricci metrics on connected sums of complex, quaternionic, and octonionic projective spaces in all dimensions.

## Key findings

- Constructed positive Ricci metrics on connected sums of projective spaces.
- Extended Perelman's techniques to broader classes of manifolds.
- Demonstrated unbounded Betti numbers for these manifolds.

## 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 arbitrary connected sums of complex projective planes. In this paper, we revisit and extend Perelman's techniques to construct positive Ricci metrics on arbitrary connected sums of complex, quaternionic, and octonionic projective spaces in every dimension.