Showing papers in "Journal of Differential Geometry in 1974"

Quaternion Kählerian manifolds

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.

Geometry of complex manifolds similar to the Calabi-Eckmann manifolds

Abstract: In [4] Calabi and Eckmann showed that the product of two odd-dimensional spheres S x S (p,q > 1) is a complex manifold. As S x S is not Kaehlerian, the fundamental 2-form Ω of the Hermitian structure is not closed. However, dΩ does have a special form on S X S in fact, S x S admits two nonvanishing vector fields which are both Killing and analytic, and whose covariant forms determine Ω. Our purpose here is to study complex manifolds whose complex structures are similar to the complex structure on S X S. In § 1 we review the geometry of the Calabi-Eckmann manifolds. In § 2 we give some elementary properties of vector fields on a Hermitian manifold, and introduce the notion of a holomorphic pair of automorphisms and of a bicontact manifold. § 3 continues the author's paper [2] on the differential geometry of principal toroidal bundles for the present case. In § 4 we discuss bicontact manifolds and, in particular, the integrable distributions of a bicontact structure on a Hermitian manifold. Finally in § 5 we give some results on the curvatures of a Hermitian manifold admitting a holomorphic pair of automorphisms.

The total absolute curvature of closed curves in Riemannian manifolds

Abstract: This paper is concerned with some extensions of the theorems of Fenchel [3], Milnor [5], Fary [2] on the total absolute curvature of closed curves in euclidean space. We obtain a theorem for closed curves in a complete simply connected riemannian n-manifold with nonpositive sectional curvature, and this leads to more precise results when the curvature is constant. Throughout this paper the summation convention for repeated indices is used, and all indices take the values 1, , n unless stated otherwise.