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By the work of Li, a compact co-Kähler manifold M is a mapping torus K ϕ , where K is a Kähler manifold and ϕ is a Hermitian isometry. We show here that there is always a finite cyclic cover M of the form M ∼ = K × S 1 , where ∼ = is equivariant diffeomorphism with respect to an action of S 1 on M and the action of S 1 on K × S 1 by translation on the second factor. Furthermore, the covering transformations act diagonally on S 1 , K and are translations on the S 1 factor. In this way, we see that, up to a finite cover, all compact co-Kähler manifolds arise as the product of a Kähler manifold and a circle. MSC classification [2010]: Primary 53C25; Secondary 53B35, 53C55, 53D05.Key words and phrases: co-Kähler manifolds, mapping tori.Let (M 2n+1 , J, ξ, η, g) be an almost contact metric manifold given by the conditionsThe authors of [CDM] use the term cosymplectic for Li's co-Kähler because they view these manifolds as odd-dimensional versions of symplectic manifoldseven as far as being a convenient setting for time-dependent mechanics [DT]. Li's characterization, however, makes clear the true underlying Kähler structure, so we have chosen to follow his terminology.

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