Compact conformally Kähler Einstein-Weyl manifolds

We give a description of compact conformally Kähler Einstein-Weyl manifolds whose Ricci tensor is Hermitian.

vector field with special Kähler-Ricci potential and consequently M = CP n or M = P(L ⊕O) where L is a holomorphic line bundle over Kähler-Einstein manifold. In Sect. 4 we prove that if M = P(L ⊕ O) then L is a holomorphic line bundle over Kähler-Einstein manifold with positive scalar curvature.

Einstein-Weyl geometry and Killing tensors
We start with some basic facts concerning Einstein-Weyl geometry. For more details see [17,15,16].
Let M be a n-dimensional manifold with a conformal structure [g] and a torsion-free affine connection D. This defines an Einstein-Weyl (E-W) structure if D preserves the conformal structure, i.e., there exists a 1-form ω on M such that Dg = ω ⊗ g (1.1) and the Ricci tensor ρ D of D satisfies the condition for some function¯ ∈ C ∞ (M). Gauduchon proved ( [7]) the fundamental theorem that if M is compact then there exists a Riemannian metric g 0 ∈ [g] for which δω 0 = 0 and g 0 is unique up to homothety. We shall call g 0 a standard metric of E-W structure (M, [g], D). Let ρ be the Ricci tensor of (M, g) and let us denote by S the Ricci endomorphism of (M, g), i.e., ρ(X, Y ) = g(X, SY ). We recall two important theorems (see [17,15]):

Theorem 1.1 A metric g and a 1-form ω determine an E-W structure if and only if there exists a function ∈ C ∞ (M) such that
Tod proved [17] that the Gauduchon metric admits a Killing vector field, more precisely he proved: Theorem 1.2 Let M be a compact E-W manifold and let g be the standard metric with the corresponding 1-form ω. Then the vector field ω dual to the form ω is a Killing vector field on M.
By τ = tr g ρ we shall denote the scalar curvature of (M, g). Compact E-W manifolds with the Gauduchon metric are Gray manifolds. To define Gray manifolds we define first a Killing tensor.
Definition A self-adjoint (1, 1) tensor on a Riemannian manifold (M, g) is called a Killing tensor if g(∇ S(X, X ), X ) = 0 for arbitrary X ∈ T M. Remark The condition g(∇ S(X, X ), X ) = 0 is equivalent to for arbitrary X, Y, Z ∈ X(M), where C denotes the cyclic sum.

Now we can give
Definition A Riemannian manifold (M, g) will be called a Gray A ⊕ C ⊥ manifold if the tensor ρ − 2τ n+2 g is a Killing tensor. In this paper, Gray A ⊕ C ⊥ manifolds will be called for short Gray manifolds or A ⊕ C ⊥ manifolds. Gray manifolds were first defined by Gray ([8]).
In the next two theorems, which are proved also in [9], we characterize compact E-W manifolds (M, g) with the Gauduchon metric as Gray ([8]) manifolds and show that the eigenvalues λ 0 , λ 1 of the Ricci tensor satisfy the equation (n − 4)λ 1 + 2λ 0 = C 0 = const which we shall use later. We sketch the proofs of the theorems. Our motivation is to use the structure of Gray manifolds to use ideas from [10], where we classified a different class of Gray manifolds. From the above theorems it follows (see [9]) ) be a compact E-W manifold, n = dimM ≥ 3, and let g be the standard metric on M. Then (M, g) is an A ⊕ C ⊥ -manifold. The manifold (M, g) is Einstein or the Ricci tensor ρ ∇ of (M, g) has exactly two eigenfunctions λ 0 ∈ C ∞ (M), λ 1 = satisfying the following conditions: In the addition λ 0 = 1 n Scal D g where Scal D g = tr g ρ D denotes the conformal scalar curvature of (M, g, D).
In the next theorem we show that conversely every Gray manifold satisfying the above conditions is in fact E-W manifold (see [9]).

Theorem 1.4 Let (M, g) be a compact
Let us assume that the Ricci tensor ρ of (M, g) has exactly two eigenfunctions λ 0 , λ 1 satisfying the conditions: Then there exists a twofold Riemannian covering (M , g ) of (M, g) and a Killing vector field ξ ∈ iso(M ) such that (M , [g ]) admits two different E-W structures with the standard metric g and the corresponding 1-forms ω ∓ = ∓ξ dual to the vector fields ∓ξ . In addition, Proof Note that τ = (n − 1)λ 1 + λ 0 and C 0 = (n − 4)λ 1 + 2λ 0 . It follows that In particular λ 0 , λ 1 ∈ C ∞ (M). Let S be the Ricci endomorphism of (M, g) and let us define the tensor T := S − λ 1 I d. Since from (1.7) we have dλ 1 = 2 n+2 dτ it follows that T is a Killing tensor with two eigenfunctions: μ = 0 and λ = λ 0 − λ 1 . Hence there exists a twofold Riemannian covering p : (M , g ) → (M, g) and a Killing vector field ξ ∈ iso(M ) (see [9], It is easy to check that with such a choice of ω Eq. (1.4) is satisfied and δω = 0. Thus (M , g , ω) defines an E-W structure and g is the standard metric for (M , [g ]). Note that (M, g , −ω) gives another E-W structure corresponding to the field −ξ .

Killing tensors
In this section, we describe the Riemannian manifold We say, that a distribution (not necessarily integrable) D is totally geodesic, if ∇ X X ∈ (D) for every X ∈ (D). Note that if (M, g) is a compact E-W manifold with the Gauduchon metric then the distribution D λ 1 is totally geodesic since is orthogonal to the distribution spanned by a Killing vector field.
We start with: Lemma 2.1 Let S be a self-adjoint tensor on (M, g) with exactly two eigenvalues λ, μ. If the distributions D λ , D μ are both umbilical, ∇λ ∈ (D μ ), ∇μ ∈ (D λ ) and the mean curvature normals ξ λ , ξ μ of the distributions D λ , D μ respectively satisfy the equations

Conformally Kähler E-W manifolds
Let g be the standard metric of (M, [g]). Now let us recall that

Proposition 3.1 Let (M, J ) be a compact complex manifold with conformal Hermitian structure [g]. Let us assume that [g] is conformally Kähler and f 2 g is a Kähler metric on (M, J ) where g is the standard metric and f
Proof Let ∇ be a Levi-Civita connection of the standard metric g and ∇ 1 be a Levi-Civita connection of the Kähler metric g 1 = f 2 g. Note that ξ is a conformal field on (M, g 1 ), L ξ g 1 = L ξ ( f 2 g) = 2ξ ln f g 1 = σ g 1 . Every conformal field on a compact Kähler manifold is Killing (see [12]), hence consequently ξ f = 0 and ξ ∈ iso(M, g 1 ). On a Kähler compact manifold every Killing vector field is holomorphic (see [14]). Thus, ξ ∈ hol(M, J ). Note that Let us recall the definition of a special Kähler-Ricci potential ( [5,3]). where (M, g, J ) is a Kähler manifold, is called a special Kähler-Ricci potential if the field X = J (∇τ ) is a Killing vector field and at every point with dτ = 0 all nonzero tangent vectors orthogonal to the fields X, J X are eigenvectors of both ∇dτ and the Ricci tensor ρ of (M, g, J ). Proof Let ρ, ρ 1 be the Ricci tensors of conformally related riemannian metrics g, g 1 = f 2 g. Then

Theorem 3.2 Let us assume that (M, [g], J ) is a compact, conformally Kähler E-W manifold with Hermitian Ricci tensor ρ D which is not conformally
We shall show that ξ has zeros on M. If ξ = 0 on M then the function φ would be defined and smooth on the whole of M. Since M is compact it would imply that there exists a point x 0 ∈ M such that dφ = 0 at x 0 . On the other hand, the eigenvalues λ 0 , λ 1 of the Ricci tensor ρ satisfy λ 0 − λ 1 = Cg(ξ, ξ ) where C = 0 is a real number. Since ξ = 0 it follows that the eigenvalues of ρ do not coincide at any point of M. In particular ρ is not J -invariant at x 0 , a contradiction, since the right hand part of (3.1) is J -invariant. It implies that ξ is a holomorphic Killing vector field with zeros and thus has a potential τ (see [11]), i.e., there exists τ ∈ C ∞ (M) such that ξ = J ∇ 1 τ . Hence, d f = −φdτ and dφ ∧ dτ = 0. It implies that dφ = αdτ . Thus, we have for arbitrary X, Y ∈ X(M): where Q = g 1 (ξ, ξ ).