We propose to investigate the conformal geometry of flag manifolds. Precisely, we focus on Riemannian metrics g on the manifold such that its conformal class contains a metric g_0 with a particular characteristic, such as being Kahler, with holonomy contained in G_2, or relaxations of these two. In some cases we relax the fact of being (globally) conformal and we only require g_0 to be locally conformal to g.Finding special metrics on the conformal class of a given metric, or even stating if for each metric there is, indeed, a special metric in its conformal class are classical problems in geometry and their treatments depend on the nature of the "special metric" in consideration. A famous problem is the Yamabe problem, for which "special" means constant scalar curvature. In this project we fix the ambient manifolds: flag manifolds. These are compact homogeneous manifolds arising from semisimple Lie groups. The description of the isotropy representation of real flag manifolds given by Patrão and San Martin in 2015 was a starting point to study invariant geometry in real flag manifolds. Their Riemannian, complex and symplectic geometry has been already considered and now starting to be understood. Still many questions regarding their geometry and topology remain to be stressed. These studies on complex flags are well known and classical. Nevertheless the conformal geometry of both has not been systematically resolved yet.The "special metrics" for us will be mostly Kahler metrics or metrics arising from a G_2 structure, for which we will require an integrability condition, such as being closed, or co-closed, or being simultaneously closed and co-closed.The proposal is to contribute to the study of conformal geometry of flag manifolds in general, by making advances in the context of locally conformal symplectic, Kahler and $G_2$. It is our purpose to provide explicit examples and attempt to obtain a classification of these structures. The Einstein condition and parallel structures for the metrics appearing in last two cases will be of particular interest.
News published in Agência FAPESP Newsletter about the scholarship: