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

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