On $\delta$-homogeneous Riemannian manifolds

We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut $\delta$-homogeneous spaces in the case of Riemannian manifolds. Every such manifold has non-negative sectional curvature. The universal covering of any $\delta$-homogeneous Riemannian manifolds is itself $\delta$-homogeneous. In turn, every simply connected Riemannian $\delta$-homogeneous manifold is a direct metric product of an Euclidean space and compact simply connected indecomposable homogeneous manifolds; all factors in this product are itself $\delta$-homogeneous. We find different characterizations of $\delta$-homogeneous Riemannian spaces, which imply that any such space is geodesic orbit (g.o.) and every normal homogeneous Riemannian manifold is $\delta$-homogeneous. The g.o. property and the $\delta$-homogeneity property are inherited by closed totally geodesic submanifolds. Then we find all possible candidates for compact simply connected indecomposable Riemannian $\delta$-homogeneous non-normal manifolds of positive Euler characteristic and a priori inequalities for parameters of the corresponding family of Riemannian $\delta$-homogeneous metrics on them (necessarily two-parametric). We prove that there are only two families of possible candidates: non-normal (generalized) flag manifolds $SO(2l+1)/U(l)$ and $Sp(l)/U(1)\cdot Sp(l-1)$, $l\geq 2$, investigated earlier by W. Ziller, H. Tamaru, D.V. Alekseevsky and A. Arvanitoyeorgos. At the end we prove that the corresponding two-parametric family of Riemannian metrics on $SO(5)/U(2)=Sp(2)/U(1)\cdot Sp(1)$ satisfying the above mentioned (strict!) inequalities, really generates $\delta$-homogeneous spaces, which are not normal and are not naturally reductive with respect to any isometry group.