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We give necessary and sufficient conditions for warped product manifolds (M, g) , of dimension ⩾ 4 , with 1 -dimensional base, and in particular, for generalized Robertson-Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R•C −C •R , formed from the curvature tensor R and the Weyl conformal curvature tensor C , is expressed by the Tachibana tensor Q(S, R) formed from the Ricci tensor S and R . We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S − α g) ⩽ 1 , for some α ∈ R , or non-quasi-Einstein.

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Introduction. A semi-Riemannian manifold (M, g), n = dim M ≥ 3, is said to be semisymmetric The semi-Riemannian manifold (M, g), n ≥ 3, satisfying (2) is called Riccisemisymmetric. There exist non-semisymmetric Ricci-semisymmetric manifolds. However, under some additional assumptions, (1) and (2) are equivalent for certain manifolds. For instance, we have the following statement.Remark 1.1. (1) and (2) are equivalent on every 3-dimensional semiRiemannian manifold as well as at all points of any semi-Riemannian manifold (M, g), of dimension ≥ 4, at which the Weyl tensor C of (M, g) vanishes (see e.g. [15, Lemma 2]). In particular, (1) and (2) are equivalent for every conformally flat manifold.It is a long standing question whether (1) and (2)

Chen (1999) established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemanian space form with arbitrary codimension. Matsumoto (to appear) dealt with similar problems for sub-manifolds in complex space forms.In this article we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in (K, μ)-contact space forms.