Abstract. We give a description of compact Einstein-Weyl manifolds in terms of Killing tensors.0. Introduction. In this paper we investigate compact Einstein-Weyl structures (M, [g], D). In the first part we consider the Killing tensors on a Riemannian manifold (M, g). We prove that if a Killing tensor S has two eigenfunctions λ, µ such that dim ker(S − λI) = 1 and µ is constant then any section ξ of the bundle D λ = ker(S − λI) such that g(ξ, ξ) = |λ − µ| is a Killing vector field on (M, g). We prove that if (M, g) is compact and simply connected then every Killing tensor field with at most two eigenvalues λ, µ at each point of M such that µ is constant and dim D λ ≤ 1 admits a Killing eigenfield ξ ∈ iso(M ) (Sξ = λξ). We also show that if the Ricci tensor of an A-manifold has at most two eigenvalues at each point then these eigenvalues have to be constant on the whole of M .In the second part we apply our results concerning Killing tensors and give a detailed description of compact Einstein-Weyl manifolds as a special kind of A⊕C ⊥ -manifolds first defined by A. Gray ([6]) (see also [1]). We show that the Ricci tensor of the standard Riemannian structure (M, g 0 ) of an Einstein-Weyl manifold (M, [g], D) can be represented as S + Λ Id T M where S is a Killing tensor and Λ is a smooth function on M . We prove that for compact simply connected manifolds there is a 1-1 correspondence between A ⊕ C ⊥ -Riemannian structures whose Ricci tensor has at most two eigenvalues at each point satisfying certain additional conditions and Einstein-Weyl structures. We also prove that if (M, [g]

We present some examples of Killing tensors and give their geometric interpretation. We give new examples of non-compact complete and compact Riemannian manifolds whose Ricci tensor satisfies the condition ∇ X (X, X) = 2 n+2 Xτ g(X, X). 0. Introduction. Killing tensors are symmetric (0, 2) tensors on a Riemannian manifold (M, g) satisfying the condition (K) ∇ X (X, X) = 0 for all X ∈ X(M) or equivalently C X,Y,Z ∇ X (Y, Z) = 0 for all X, Y, Z ∈ X(M) where X(M) denotes the space of all local vector fields on M , C denotes the cyclic sum and ∇ denotes the Levi-Civita connection of (M, g). The condition (K) is a generalization of the condition ∇ = 0. Another generalization of this condition is ∇ X (Y, Z) = ∇ Y (X, Z), which gives the class of Codazzi tensors. The Codazzi tensors are quite frequently used in Riemannian geometry. For example the second fundamental form of any hypersurface immersed in a Euclidean space is a Codazzi tensor. On the other hand it is difficult to find general examples of Killing tensors in the literature. It is only known that a Ricci tensor of any naturally reductive homogeneous space (and more generally of any D'Atri space) has this property. The aim of the present paper is to show that Killing tensors appear quite naturally in Riemannian geometry. We prove that on every warped product M 0 × f 1 M 1 ×. .. × f k M k there exists a Killing tensor Φ(X, Y) = g(SX, Y) such that the functions λ 0 = µ ∈ R and λ i = µ + C i f 2 i for i > 1 are eigenfunctions of S for any µ ∈ R and any real constants C i ∈ R − {0}. Conversely, let Φ(X, Y) = g(SX, Y) be a Killing tensor with an integrable almost product structure given by its eigendistributions and with eigenvalues µ, λ 1 ,. .. , λ k such that µ ∈ R is constant and j>0 D j ⊂ ker dλ i (i.e. ∇λ i ∈

Abstract. The aim of this paper is to give a characterization of 3-K-contact and quasi 3-K-contact manifolds.

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