Quaternion Kählerian manifolds

TLDR: Alekseevskii et al. as mentioned in this paper studied quaternion Kahlerian manifolds by using tensor calculus and obtained many interesting results. But they did not define a manifold as a Riemannian manifold which admits a bundle V of tensors of type (1, 1) having some properties.

Abstract: A quaternion Kahlerian manifold is defined as a Riemannian manifold whose holonomy group is a subgroup of Sp(m) Sp(l) . Recently, several authors (Alekseevskii [1], [2], Gray [3], Ishihara [4], Ishihara and Konishi [5], Krainse [6] and Wolf [10]) have studied quaternion Kahlerian manifolds and obtained many interesting results. In the present paper, we shall study those manifolds by using tensor calculus. To do so, it is rather convinient to define a quaternion Kahlerian manifold as a Riemannian manifold which admits a bundle V of tensors of type (1,1) having some properties. The bundle V is 3-dimensional as a vector bundle and admits an algebraic structure which is isomorphic to that of pure imaginary quaternions. In § 1, we define quaternion Kahlerian manifolds in our fashion and give some results proved in [6]. § 2 is devoted to the establishment of some formulas required in the following sections. In § 3, it is proved among some other theorems that any quaternion Kahlerian manifold is an Einstein space (Alekseevskii [1]). We prove in § 4 that a quaternion Kahlerian manifold, which is of constant curvature or conformally flat, is of zero curvature, if the manifold is of dimension > 8 . In §5, we define β-sectional curvatures and determine the form of the curvature tensor of a quaternion Kahlerian manifold when it has constant β-sectional curvature (See Alekseevskii [1]). Manifolds, mappings and geometric objects under discussion are assumed to be differentiable and of class C°°. The indices h, i, /, k, I, p, q, r, s, t, u, v run over the range {1, , n}, and the summation convention will be used with respect to this system of indices.