Matter couplings in N = 2 supergravity

Quaternionic Kähler Manifolds.

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The conjectures on conformal transformations of Riemannian manifolds

In [1], J. Eells and L. Lemaire introduced the notion of a k-harmonic map. In this paper we study the case k = 2, derive the first and second variational formulas of the 2-harmonic maps, give nontrivial examples of 2-harmonic maps and give proofs of nonexistence theorems of stable 2-harmonic maps.

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2-harmonic maps and their first and second variational formulas

Introduction and theorems. In this paper we shall always deal with connected Riemannian manifolds with positive definite metric, and suppose that manifolds and quantities are differentiable of class C'. Let M be an n-dimensional Riemannian manifold with metric tensor field (1) g,,. We call a nonconstant scalar field p in M a concircular scalar field, or simply a concircularfield, if it satisfies the equation

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Complete Riemannian manifolds and some vector fields

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.

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Quaternion Kählerian manifolds

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Homogeneous Riemannian spaces of four dimensions.

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Almost Hermitian structure on $S^6$

ON INTEGRABILITY CONDITIONS OF A STRUCTURE f SATISFYING f3+f=0

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Shigeru Ishihara

Papers

Quaternion Kählerian manifolds

Homogeneous Riemannian spaces of four dimensions.

Almost Hermitian structure on $S^6$

ON INTEGRABILITY CONDITIONS OF A STRUCTURE f SATISFYING f3+f=0

Holomorphically projective changes and their groups in an almost complex manifold