The Ricci curvature and the normalized Ricci flow on the Stiefel manifolds $\operatorname{SO}(n)/\operatorname{SO}(n-2)$

TL;DR

This paper proves that the normalized Ricci flow on Stiefel manifolds preserves and enhances positive Ricci curvature for invariant metrics, with all metrics evolving into positive Ricci curvature in finite time, and analyzes the flow's invariant sets.

Contribution

It establishes the preservation and improvement of Ricci curvature under the normalized Ricci flow on Stiefel manifolds and characterizes invariant sets related to positive Ricci curvature.

Findings

Normalized Ricci flow preserves positive Ricci curvature.

All metrics with mixed Ricci curvature evolve into positive Ricci curvature.

Existence of invariant sets with positive Ricci curvature parameters.

Abstract

We proved that on every Stiefel manifold $V_2\mathbb{R}^n\cong \operatorname{SO}(n)/\operatorname{SO}(n-2)$ with $n\ge 3$ the normalized Ricci flow preserves the positivity of the Ricci curvature of invariant Riemannian metrics with positive Ricci curvature. Moreover, the normalized Ricci flow evolves all metrics with mixed Ricci curvature into metrics with positive Ricci curvature in finite time. From the point of view of the theory of dynamical systems we proved that for every invariant set~$\Sigma$ of the normalized Ricci flow on~$V_2\mathbb{R}^n$ defined as $x_1^{n-2}x_2^{n-2}x_3=c$, $c>0$, there exists a smaller invariant set $\Sigma\cap \mathscr{R}_{+}$ for every $n\ge 3$, where~$\mathscr{R}_{+}$ is the domain in $\mathbb{R}_{+}^3$ responsible for parameters $x_1, x_2, x_3>0$ of invariant Riemannian metrics on~$V_2\mathbb{R}^n$ admitting positive Ricci curvature.