The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.

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We determine an orthogonal decomposition of the vector space of some curvature tensors on a co-Hermitian real vector space, in irreducible components with respect to the natural induced representation of c U(n)xl.One of the components is used to introduce a Bochner curvature tensor on a class of almost co-Hermitian manifolds (or almost contact metric manifolds), called C(ά)-manifolds, containing e.g. co-Kahlerian, Sasakian and Kenmotsu manifolds. Other applications of the decomposition are given.

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Abstract.A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.1. Introduction. Let (V, g) be an n-dimensional real vector space with positive definite inner product g and denote by 61 (V) the subspace of V* <8> V* <8> V* <8> V* consisting of all tensors having the same symmetries as the curvature tensor of a Riemannian manifold, including the first Bianchi identity. In a well-known paper [21] Singer and Thorpe considered 61(F) (in particular for n = 4) and gave a geometrical useful description of the splitting of 61(F) under the action of &(n) into three components. This was also studied by Nomizu [18] for generalized curvature tensor fields.A similar decomposition was given in [16], [17] and [22] when V is a 2«-dimensional real vector space endowed with a complex structure J compatible with a positive definite inner product g and for the subspace %(V) of 61(F) consisting of