Killing tensors and Einstein-Weyl geometry
Włodzimierz Jelonek
1999
Colloquium Mathematicum
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
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... at 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], D) is a compact Einstein-Weyl manifold with dim M ≥ 4 which is not conformally Einstein then the conformal scalar curvature s D of (M, [g], D) is nonnegative and that the center of the Lie algebra of the isometry group of the standard Riemannian structure (M, g 0 ) of (M, [g], D) is nontrivial. Our results rely on some results of P. Gauduchon [3] and H. Pedersen and A. Swann ([9], [10]).
doi:10.4064/cm-81-1-5-19
fatcat:ex4u5a7zmffuvgo2kf2alnv4n4