On the curvature ODE associated to the Ricci flow

Abstract

In the vector space of algebraic curvature operators we study the reaction ODE $$\begin{aligned} \frac{dR}{dt} = R^2+R^{\#}= Q(R) \end{aligned}$$ which is associated to the evolution equation of the Riemann curvature operator along the Ricci flow. More precisely, we give a partial classification of the zeros of this ODE up to suitable normalization and analyze the stability of a special class of zeros of the same. In particular, we show that the ODE is unstable near the curvature operators of the Riemannian product spaces \(M \times \mathbb {R}^k, k \ge 0\) where \(M\) is an Einstein (locally) symmetric space of compact type and not a spherical space form when \(k=0\).