  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.
