2006/11/20 by V. N. Berestovskiĭ, V. N. Berestovskii, Berestovskii, V. N. +2
Mathematics · Physics and Astronomy · #53C20 (primary) #53C25 #53C35 (secondary) #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C20 #msc:53C25 #msc:53C35
paper · pdf · doi:10.48550/arxiv.math/0611557
40 pages, some results are strengthened, new references are added
openalex publication_date 2006/11/20 · arxiv created 2007/02/09 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut δ-homogeneous spaces in the case of Riemannian manifolds. Every such manifold has non-negative sectional curvature. The universal covering of any δ-homogeneous Riemannian manifolds is itself δ-homogeneous. In turn, every simply connected Riemannian δ-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 δ-homogeneous. We find different characterizations of δ-homogeneous Riemannian spaces, which imply that any such space is geodesic orbit (g.o.) and every normal homogeneous Riemannian manifold is δ-homogeneous. The g.o. property and the δ-homogeneity property are inherited by closed totally geodesic submanifolds. Then we find all possible candidates for compact simply connected indecomposable Riemannian δ-homogeneous non-normal manifolds of positive Euler characteristic and a priori inequalities for parameters of the corresponding family of Riemannian δ-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)⋅ Sp(l-1), l≥ 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)⋅ Sp(1) satisfying the above mentioned (strict!) inequalities, really generates δ-homogeneous spaces, which are not normal and are not naturally reductive with respect to any isometry group.