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. 
V.N. Berestovskii (Omsk Branch of Sobolev Institute of Mathematics SD   RAS), Yu.G. Nikonorov (Rubtsovsk Industrial Institute)