Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic

Abstract

The authors give a short survey of previous results on δ-homogeneous 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 δ-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) = CP 2l−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.2000 Mathematical Subject Classification: 53C20 (primary), 53C25, 53C35 (secondary).

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

“…In fact we use in this paper another notation, namely, c 2 := 1, t := b 2 , s := a 2 , which coincides with the notation in papers [28], [29], [27]. Then, summing up and comparing Tables 1 and 2, applying the notation from Section 1, and adding at the end the mentioned result from paper [4], we obtain the following statement. Theorem 1.…”

“…In papers [7,8,4] the authors studied a class of generalized normal homogeneous Riemannian manifolds (δ-homogeneous Riemannian manifolds, in other terms).…”

“…It follows from Theorem 2 that any (G-)normal homogeneous Riemannian manifold [13] is also (G-)δ-homogeneous [7]. The converse statement is not true for some spaces [7,4]. The existence of such spaces is always connected with unusual geometric properties of adjoint representations of corresponding Lie groups G [7,4].…”

“…The converse statement is not true for some spaces [7,4]. The existence of such spaces is always connected with unusual geometric properties of adjoint representations of corresponding Lie groups G [7,4]. The complete classification of such simply connected Riemannian manifolds with positive Euler characteristic, indecomposable into direct metric product, is given in paper [4].…”

“…The existence of such spaces is always connected with unusual geometric properties of adjoint representations of corresponding Lie groups G [7,4]. The complete classification of such simply connected Riemannian manifolds with positive Euler characteristic, indecomposable into direct metric product, is given in paper [4]. These are exactly all complex projective spaces with invariant Riemannian metrics (CP 2n+1 = Sp(n + 1)/(U(1) × Sp(n)), ν t ), for n ≥ 1, 1 2 < t < 1, considered also in this paper, as well as the spaces homothetic to them.…”