Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic

Abstract

The authors give a short survey of previous results on generalized normal homogeneous ( δ-homogeneous, in other terms) Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with nonnegative 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 generalized normal homogeneous Riemannian manifolds with positive Euler characteristic, which are not normal homogeneous, consists exactly of all generalized flag manifolds Sp ( l ) / U ( 1 ) ⋅ Sp ( l − 1 ) = C P 2 l − 1 , l ⩾ 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 ⩾ 2 . Some unsolved questions are suggested.