Ricci flow and curvature on the variety of flags on the two dimensional projective space over the complexes, quaternions and the octonions

TL;DR

This paper investigates how Ricci flow affects the curvature properties of homogeneous metrics on flag varieties over complex, quaternionic, and octonionic projective spaces, showing that positive curvature can evolve into indefinite or negative curvature.

Contribution

It demonstrates that Ricci flow can alter the curvature signature of homogeneous metrics on these flag varieties, revealing new dynamical behaviors of curvature under Ricci flow.

Findings

Ricci flow can change positive sectional curvature to indefinite or negative curvature.

Positive definite Ricci tensor can evolve into indefinite signature.

Curvature properties are dynamically altered by Ricci flow on these spaces.

Abstract

For homogeneous metrics on the spaces of the title it is shown that the Ricci flow can move a metric of stricly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with indefinite signature.