The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow

2015/09/30 by N. A. Abiev, Nurlan Abievich Abiev, Yu. G. Nikonorov +1 · 16 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Curvature of Riemannian manifolds #Differential geometry #Geometric Analysis and Curvature Flows #Invariant (physics) #Nonlinear Partial Differential Equations #Ricci curvature #Ricci flow #Sectional curvature #Yamabe flow #math.DG #math.DS #msc:34C05 #msc:37C10 #msc:53C30 #msc:53C44

paper · pdf · doi:10.1007/s10455-016-9502-8

published in Annals of Global Analysis and Geometry 50(1), 65-84 (Springer Science+Business Media) · 20 pages, 10 figures, proofs simplified, new results added, references updated

arxiv created 2016/02/16 · openalex publication_date 2016/02/26 · openalex created_date 2016/06/24 · arxiv updated 2020/05/19 · openalex updated_date 2026/08/05

Abstract

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces SU(3)/T_max, Sp(3)/Sp(1)× Sp(1)× Sp(1), and F₄/Spin(8). We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, we prove that for the spaces Sp(3)/Sp(1)× Sp(1)× Sp(1) and F₄/Spin(8), the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci curvature. We also get similar results for some more general homogeneous spaces.

Citations

• Non-singular solutions of the Ricci flow on three-manifolds
• Three-manifolds with positive Ricci curvature
• Compact Homogeneous Riemannian Manifolds with Strictly Positive Curvature
• An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures
• Classification of generalized Wallach spaces
• Einstein Manifolds