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

Nurlan A. Abiev; Nurlan A. Abiev

doi:10.3934/era.2025084

Electronic Research Archive

2025, Volume 33, Issue 3: 1858-1874. doi: 10.3934/era.2025084

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

Nurlan A. Abiev ^,

Institute of Mathematics of NAS of the Kyrgyz Republic, Bishkek 720071, Kyrgyz Republic

Received: 30 November 2024 Revised: 05 March 2025 Accepted: 12 March 2025 Published: 31 March 2025

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.
- Stiefel manifold,
- Riemannian metric,
- normalized Ricci flow,
- Ricci curvature,
- dynamical system,
- invariant set,
- singular point
Citation: Nurlan A. Abiev. The Ricci curvature and the normalized Ricci flow on the Stiefel manifolds $ \operatorname{SO}(n)/\operatorname{SO}(n-2) $[J]. Electronic Research Archive, 2025, 33(3): 1858-1874. doi: 10.3934/era.2025084

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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.

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