The complete dynamics description of positively curved metrics in the Wallach flag manifold $\mathrm{SU}(3)/\mathrm{T}^2$

TL;DR

This paper characterizes regions within the parameter space of invariant metrics on the Wallach flag manifold $ ext{SU}(3)/ ext{T}^2$ that exhibit various positive curvature properties and analyzes their evolution under the Ricci flow.

Contribution

It provides a complete description of the curvature regions in the parameter space and studies their dynamics under the projected Ricci flow, including intermediate curvature notions.

Findings

Identifies all regions with positive sectional and scalar curvature.

Analyzes the behavior of these regions under Ricci flow, including sign preservation and escape.

Extends results to positive intermediate Ricci curvature on fiber bundles.

Abstract

The family of invariant Riemannian manifolds in the Wallach flag manifold $\mathrm{SU}(3)/\mathrm{T}^2$ is described by three parameters $(x,y,z)$ of positive real numbers. By restricting such a family of metrics in the \emph{tetrahedron} $\cal{T}:= x+y+z = 1$, in this paper, we describe all regions $\cal R \subset \cal T$ admitting metrics with curvature properties varying from positive sectional curvature to positive scalar curvature, including positive intermediate curvature notion's. We study the dynamics of such regions under the \emph{projected Ricci flow} in the plane $(x,y)$, concluding sign curvature maintenance and escaping. In addition, we obtain some results for positive intermediate Ricci curvature for a path of metrics on fiber bundles over $\mathrm{SU}(3)/\mathrm{T}^2$, further studying its evolution under the Ricci flow on the base.