The classification of $\delta$-homogeneous Riemannian manifolds with positive Euler characteristic

TL;DR

This paper classifies certain $oldsymbol{ ext{delta}}$-homogeneous Riemannian manifolds with positive Euler characteristic, revealing they are specific generalized flag manifolds with particular curvature properties, and discusses their unique geometric features.

Contribution

It identifies all compact simply connected indecomposable $oldsymbol{ ext{delta}}$-homogeneous manifolds with positive Euler characteristic that are not normal homogeneous, as specific flag manifolds with positive curvature metrics.

Findings

Classified $oldsymbol{ ext{delta}}$-homogeneous manifolds with positive Euler characteristic.

Identified these manifolds as certain generalized flag manifolds $ ext{Sp}(l)/U(1) ext{Sp}(l-1)$.

Established curvature pinching constants in the interval $(1/16, 1/4)$ for these manifolds.

Abstract

The authors give a short survey of previous results on $\delta$-homogeneous Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these results, they prove that the family of all compact simply connected indecomposable $\delta$-homogeneous Riemannian manifolds with positive Euler characteristic, which are not normal homogeneous, consists exactly of all generalized flag manifolds $Sp(l)/U(1)\cdot Sp(l-1)=\mathbb{C}P^{2l-1}$, $l\geq 2$, supplied with invariant Riemannian metrics of positive sectional curvature with the pinching constants (the ratio of the minimal sectional curvature to the maximal one) in the open interval $(1/16, 1/4)$. This implies very unusual geometric properties of the adjoint representation of $Sp(l)$, $l\geq 2$. Some unsolved questions are suggested.