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

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 tetrahedron $\mathcal{T}:= x+y+z = 1$, we show how to describe all regions $\mathcal R \subset \mathcal T$ 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 projected Ricci flow in the plane $(x,y)$, concluding sign curvature maintenance and escaping. We stress how this approach can be generalized to several other homogeneous spaces and can be helpful to discuss the moduli space of bundles associated with the principal bundle $\mathrm{T}^2\hookrightarrow \mathrm{SU}(3) \rightarrow \mathrm{SU}(3)/\mathrm{T}^2$.

This work is done in collaboration with Lino Grama, Ricardo M. Martins and Douglas D. Novaes

classical analysis and ODEsdifferential geometrymetric geometry

Audience: researchers in the topic

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