Curvature cones and the Ricci flow

TL;DR

This survey discusses curvature conditions preserved by Ricci flow, highlighting known examples, their geometric applications, recent restrictions, and open questions in the field.

Contribution

It reviews known preserved curvature conditions under Ricci flow and presents recent results on restrictions and open problems.

Findings

Examples of preserved curvature conditions and their geometric implications

Recent restrictions on general preserved conditions

Open questions about curvature cones and Ricci flow

Abstract

This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points. First we describe the known examples of preserved curvature con-ditions and how they have been used to derive geometric results, in particular sphere theorems. We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters. The Ricci flow is the following evolution equation: