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

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.