标题： 里奇曲率和施蒂费尔流形上的归一化里奇流$\operatorname{SO}(n)/\operatorname{SO}(n-2)$ 显示英文标题

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

摘要： 我们证明了在每个Stiefel流形$V_2\mathbb{R}^n\cong \operatorname{SO}(n)/\operatorname{SO}(n-2)$上，当$n\ge 3$时，归一化Ricci流保持具有正Ricci曲率的不变黎曼度量的Ricci曲率的正性。 此外，归一化Ricci流在有限时间内将具有混合Ricci曲率的度量演化为具有正Ricci曲率的度量。 从动力系统理论的角度来看，我们证明了对于定义在~$V_2\mathbb{R}^n$上的归一化Ricci流的每一个不变集~$\Sigma$（其中~$x_1^{n-2}x_2^{n-2}x_3=c$，~$c>0$），对于每个~$n\ge 3$，都存在一个更小的不变集~$\Sigma\cap \mathscr{R}_{+}$，其中~$\mathscr{R}_{+}$是~$\mathbb{R}_{+}^3$中负责~$V_2\mathbb{R}^n$上不变黎曼度量参数~$x_1, x_2, x_3>0$的区域，并且这些度量允许正Ricci曲率。 显示英文摘要

摘要： 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.