Skew Killing spinors in four dimensions

This paper is devoted to the classification of 4-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $\psi$ is a spinor that satisfies the equation $\nabla$X$\psi$ = AX $\times$ $\psi$ with a skew-symmetric endomorphism A. We consider the degenerate case, where the rank of A is at most two everywhere and the non-degenerate case, where the rank of A is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product R x N with N having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to S 2 x R 2 occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.


Introduction
Let (M n , g) be an n-dimensional Riemannian spin manifold. A generalised Killing spinor on M is a section ψ of the spinor bundle ΣM of M satisfying the overdetermined differential equation ∇ X ψ = AX · ψ for some symmetric endomorphism field A of T M . Here and as usual, "·" denotes the Clifford multiplication on ΣM . Numerous papers have been devoted to the classification of Riemannian spin manifolds carrying such spinors. Several results have been obtained for particular A but it is still an open problem to get a complete classification for general A. Let us quote some of these results. First, recall that when A is the zero tensor field, that is, the corresponding spinor is parallel, then McK. Wang [22] showed that such manifolds can be characterised by their holonomy groups which can be read off the Berger classification. The case where A is a nonzero real multiple of the identity is that of classical real Killing spinors. It was shown by C. Bär [2] that real Killing spinors correspond to parallel spinors on the (irreducible) cone over the manifold, to which then McK. Wang's result applies. Furthermore, in dimension n ≤ 8, there are several results on a classification up to isometry [5,16]. When the tensor A is parallel [18], or a Codazzi tensor [4] or both A and g are analytic [1] (see also [8]), it is shown that the manifold M is isometrically embedded into another spin manifold of dimension n + 1 carrying a parallel spinor and that the tensor A is the half of the second fundamental form of the immersion. We also cite the partial classification of generalised Killing spinors on the round sphere [21,19] and on 4-dimensional Einstein manifolds of positive scalar curvature [20] where in some cases the generalised Killing spinor turns out to be a Killing spinor.
In this paper, we are interested in an equation dual to the generalised Killing one, which we call skew Killing spinor equation. More precisely, on a given Riemannian spin manifold (M n , g), a spinor field ψ is called a skew Killing spinor if it satisfies for some skew-symmetric endomorphism field A of T M the differential equation for all X ∈ T M . This equation was originally defined in [14]. Each skew Killing spinor is a parallel section with respect to the modified metric connection ∇ − A ⊗ Id, in particular it has constant length. Moreover, for a given skew symmetric endomorphism field A of T M , the space of skew Killing spinors is a complex vector space of dimension at most rk C (ΣM ) = 2 [n/2] .
Very few examples of Riemannian spin manifolds (M n , g) carrying skew Killing spinors are known for which A = 0. For 2-dimensional manifolds, apart from R 2 or quotients thereof with trivial spin structure, only the round sphere of constant curvature can carry such spinors and in that case they correspond to restrictions of Killing spinors from S 3 onto totally geodesic S 2 [14]. In that case, the tensor A coincides with the standard complex structure J induced by the conformal class of S 2 or with −J depending on the sign of the Killing constant chosen on S 3 . Each skew Killing spinor on S 2 immediately gives rise to a three-dimensional example, namely to a skew Killing spinor on S 2 × R, where A = ±J on S 2 is trivially extended to the R-factor. More generally, for a manifold of dimension n = 3 the following is known [14,Prop. 4.3]. If M 3 admits a skew Killing spinor ψ, then, locally, ψ can be transformed into a parallel spinor by a suitable conformal change of the metric. In particular, M 3 is locally conformally flat. If, in addition, M 3 is simply-connected, then this conformal change is defined globally. Conversely, if (M 3 , g) admits a nonzero parallel spinor, then for any conformal change of g, there exists a skew Killing spinor with respect to the new metric. See Section 4.1 for more detailed information.
Obvious examples in four dimensions can be obtained as products N × R, where N is a threedimensional manifold admitting a skew Killing spinor, see Example 4.1. A special case of this construction is the product S 2 ×R 2 , see Example 4.2. For each of the endomorphisms A ± := ±J ⊕0, this manifold admits the maximal number of skew Killing spinors.
The main purpose of this work is to establish a classification result when the dimension of M is four. Note that the pointwise rank of A is either zero, two or four. We will split the classification into two parts. In Section 4 we will study the degenerate case, where the rank of A is at most two everywhere. In Section 5 we will consider the case where rk(A) = 4 on all of M . Before we start the classification, we determine the general integrability conditions in arbitrary dimensions arising from the existence of a skew Killing spinor, see Section 2. In Sections 3 and 4, we specify these conditions to four dimensions, especially to the degenerate case. We use that the spinor bundle ΣM splits into the eigenspaces Σ + M and Σ − M of the volume form and the bundle of two-forms splits into those of self-dual and of anti-self-dual forms, which act on Σ ± M . We also adapt some techniques used in [20] but for a skew-symmetric endomorphism A. We use the integrability conditions to achieve the following classification result in case that the Killing map is degenerate everywhere.
Theorem A. Let (M 4 , g) be a connected Riemannian spin manifold carrying a skew Killing spinor ψ, where the rank of the corresponding skew-symmetric tensor field A is at most two everywhere. Then either ψ is parallel on M or, around every point of M , we have a local Riemannian splitting R × N with N having a skew Killing spinor. If, in addition, the length of the summand ψ + in the decomposition ψ = ψ + + ψ − ∈ Σ + M ⊕ Σ − M is not constant, then we are in the second case with N = R × S 2 , that is, (M, g) is a local Riemannian product S 2 × R 2 around every point.
For a more detailed formulation see Theorem 4.13, where we also discuss the global structure of (M, g) if M is complete.
Let us turn to the case where the Killing map is non-degenerate everywhere. In Section 5.1 we will prove that, essentially, the existence of a skew Killing spinor ψ with non-degenerate Killing map A is equivalent to the existence of a Killing vector field η and an almost complex structure J satisfying certain conditions, see Proposition 5.1 for a detailed formulation. The spinor ψ and the data η and J are related by the equations J(X) · ψ − = iX · ψ − and g(η, X) = X · ψ + , ψ − /|ψ| 2 for all X ∈ T M .
In Section 5.2, we consider the special case where Aη is parallel to Jη. Then AJ = JA holds and J is integrable, see Remark 5.3. Manifolds with skew Killing spinors satisfying these conditions are related to doubly warped products. A doubly warped product is a Riemannian manifold (M, g) of the form (I ×M , dt 2 ⊕ρ(t) 2ĝη ⊕σ(t) 2ĝη ⊥ ), where (M ,ĝ) is a Riemannian manifold with unit Killing vector fieldη, andĝη,ĝη⊥ are the components of the metricĝ along Rη andη ⊥ , respectively, I ⊂ R is an open interval and ρ, σ: I → R are smooth positive functions on I. Locally, doubly warped products can be equivalently described as local DWP-structures, see the appendix. OnM , we define a functionτ by∇ Xη =τ ·Ĵ(X) for X ∈η ⊥ , whereĴ is a fixed Hermitian structure on η ⊥ . Locally, (M ,ĝ) is a Riemannian submersion over a two-dimensional base manifold B. LetK denote the Gaussian curvature of B. We obtain the following result, see Theorem 5.5 and Corollary 5.8.
Theorem B. Let (M, g) admit a skew Killing spinor such that Aη||Jη and |η| ∈ {0, 1/2} everywhere. Then M is locally isometric to a doubly warped product for which the dataK andτ are constant and ρ and σ satisfy the differential equations Conversely, if M is isometric to a simply-connected doubly warped product for which the dataK andτ are constant and ρ and σ satisfy the above differential equations, then (M, g) admits a skew Killing spinor such that Aη||Jη.
The differential equations in Theorem B can be locally solved and one obtains explicit formulas for the doubly warped product. Let us finally mention that the skew Killing spinors on M = I ×M are related to quasi Killing spinors in the sense of [10] onM , see Remark 5.10.
The Hodge star operator satisfies * 2 = (−1) p(n−p) on p-forms and has the following useful properties for any vector field X. Recall also that the Clifford multiplication between a vector field X and a differential p-form ω is defined as from which the identity X · Y · +Y · X· = −2g(X, Y ) follows for any vector fields X and Y.
From now on, we assume M to be spin with fixed spin structure. In that case, there exists a Hermitian vector bundle ΣM → M , called the spinor bundle, on which the tangent bundle T M acts by Clifford multiplication, T M ⊗ ΣM → ΣM ; X ⊗ ψ → X · ψ. We will write XY · ψ instead of X · Y · ψ. Recall that a real p-form also acts by Clifford multiplication in a formally self-or skew-adjoint way according to its degree: for any p-form ω and any spinors ϕ, ψ, we have The Levi-Civita connection ∇ on M defines a metric connection, also denoted by ∇, on ΣM with respect to the Hermitian product · , · and that preserves Clifford multiplication. In other words, for all X, Y ∈ Γ(T M ), the rules the curvature tensor associated with the connection ∇, the spinorial Ricci identity states that, for all ψ and X, see e.g. [5,Eq. 1.13].
In the following, we will assume the manifold M to carry a skew-Killing spinor field ψ with corresponding skew-symmetric endomorphism A. We make A into a 2-form via the metric g, that is, we consider (X, Y ) → g(AX, Y ), which we still denote by A. In a pointwise orthonormal basis {e i } i=1,···,n of T M , we have A = 1 2 n j=1 e j ∧ Ae j (mind the factor 1 2 ). In particular, Clifford multiplication of any spinor field ψ by A is given by In the next proposition, we compute the curvature data arising from the existence of such a spinor. These integrability equations will play a crucial role for the classification in the 4-dimensional case.
Proposition 2.1 Let ψ be any solution of (1) on a spin manifold (M n , g) for some skew-symmetric endomorphism field A of T M . Then the following identities hold for X, Y ∈ Γ(T M ) where d is the exterior derivative and δ is the codifferential w.r.t. the metric g.
Proof: We derive (1) and take suitable traces of the identities obtained. First, if x ∈ M and X, Y ∈ Γ(T M ) such that ∇X = ∇Y = 0 at x, then which is the first identity.
Next we fix a local orthonormal basis of T M , which we denote by (e j ) 1≤j≤n . Using the spinorial Ricci formula (4) and the identities (3), we compute Now we compute each term separately. First, n j=1 e j · (∇ X A)(e j ) · ψ = 2∇ X A · ψ by (5), where we see ∇ X A as a 2-form on M . The second sum can be computed in terms of the exterior and the covariant derivatives of A. Namely It remains to notice that, by Equations (3), we have n j=1 e j · (Ae j ∧ AX) · ψ = n j=1 e j · Ae j · AX · ψ + n j=1 g(Ae j , AX)e j · ψ This shows the second equation.
To obtain the scalar curvature, we trace the spinorial Ricci identity. Given a local orthonormal basis (e j ) 1≤j≤n of T M , we write which is the last identity. Here, we use the the identity n j=1 e j ∧ (e j ω) = pω, which holds for any p-form ω.

The vector fields η and ξ in four dimensions
In this section, we consider a 4-dimensional spin manifold (M, g) that carries a skew Killing spinor. On spin manifolds of even dimension 2m, the complex volume form (vol g ) C := i m e 1 · e 2 . . . · e 2m , where (e j ) j=1,···,2m is an arbitrary orthonormal frame, splits the spinor bundle into two orthogonal subbundles that correspond to the eigenvalues ±1 of (vol g ) C . Hence, on our four-dimensional manifold (M, g), we have ΣM = Σ + M ⊕ Σ − M , where The spaces Σ ± M are preserved by the connection ∇ of the spinor bundle and are interchanged by Clifford multiplication by tangent vectors. According to this decomposition, we write any spinor field ψ as ψ = ψ + + ψ − and we setψ := ψ + − ψ − . Recall now that differential forms act on the spinor bundle ΣM as follows: for any differential p-form ω on M and ψ ∈ Γ(ΣM ) ω · ψ = * ω ·ψ for p = 1, 2 and ω · ψ = −( * ω) ·ψ for p = 3, 4.
We collect some properties of η and ξ that will be used later on.
Proof: Differentiating the function |ψ − | 2 along any vector field X ∈ T M gives This proves 1. To prove 2, we consider two vector fields X and Y that can be assumed to be parallel at some point x ∈ M to compute which yields the first part of 3. The divergence of η is clearly zero by 2 and the fact that A is skew-symmetric. Finally, which together with 1 gives 4. The open sets M 0 and M 1 are dense in {p ∈ M | A p = 0}. Indeed, if, e.g., ψ − vanishes on some open set U ⊂ {p ∈ M | A p = 0}, then so does its covariant derivative and therefore AX · ψ + = 0 on U . Hence A = 0 on U , which contradicts the assumption on A.
With the notation introduced above, we have We define also the set

The degenerate case
In this section, we assume that rk(A) ≤ 2 everywhere on M 4 , which is equivalent to suppose that the kernel of A is at every point either 4-or 2-dimensional. Then AX ∧ A = 0 for all X ∈ T M . In particular, dA = 0 on M ′′ by Lemma 3.1.
Let us prove the above statement. Recall that the spinor bundle of M = N × R is given by ΣM = ΣN ⊕ ΣN and the Clifford multiplication on M is related to the one on N by [3] where ∂ t is the unit vector field on R and X ∈ T N . Now we set ψ := ϕ + ∂ t · ϕ according to the above decomposition. Let A denote the Killing map associated with ψ. Then we can easily check that ∇ ∂t ψ = 0 and, for X ∈ T N , Hence ψ is a skew-Killing spinor on M . The vector field ξ in this example is just −∂ t which is parallel. Since |∂ t | = 1, we have |ψ + | = |ψ − |.
Let us recall at this point, what is known about three-dimensional manifolds with skew Killing spinors. As already mentioned in the introduction, each skew Killing spinor on S 2 immediately gives rise to a three-dimensional example, namely to a skew Killing spinor on S 2 × R. Furthermore, if dim N = 3 and if (N, g) admits a skew Killing spinor ψ, then N is locally conformally flat [14,Prop. 4.3]. Indeed, locally, there exists a function u such that ψ transforms into a parallel spinorψ with respect to the metricḡ := e 2u g and three-dimensional Riemannian manifolds with a non-trivial parallel spinor field are flat. If N is simply-connected, then u is globally defined. In the latter case the metricḡ is not necessarily complete even if (N, g) is. Conversely, if (N, g) admits a nonzero parallel spinor, then for any conformal change of the metric on the manifold N there exists a skew Killing spinor with respect to the new metric. We conclude this overview with the flat case N = R 3 . If ψ = 0 is a solution of (1) on N = R 3 endowed with the flat metric, then A = 0 and ψ is a parallel spinor field. Indeed, as mentioned above, there exists a globally defined function u on R 3 such that metricḡ := e 2u g admits a parallel spinor. Hence,ḡ is also flat. In particular, the scalar curvatureS vanishes. On the other hand,S = 8e −2u e −u/2 ∆e u/2 sinceḡ arises by conformal change from the flat metric g. Thus ∆(e u/2 ) = 0, that is, e u/2 is a harmonic function on R 3 . But since e u/2 ≥ 0, Liouville's theorem implies that e u/2 -and so u itself -is constant. This shows A = 0.
For each of these endomorphisms, the space of skew Killing spinors is four-dimensional. It can be spanned by elements with non-vanishing Aη and it also can be spanned by elements for which Aη = 0 holds.
Let us prove this statement. The spinor bundle of S 2 × R 2 is pointwise given by Σ(S 2 × R 2 ) = ΣS 2 ⊗ ΣR 2 and the Clifford multiplication on for X ∈ T S 2 and Y ∈ T R 2 . Now, we consider on S 2 a skew Killing spinor ϕ, corresponding to the standard complex structure J, and a parallel spinor σ in Σ + (R 2 ) of norm 1. The spinor field ψ := ϕ ⊗ σ is clearly a skew-Killing spinor, since in the S 2 -direction we have and ∇ Y ψ = 0 in the R 2 -direction. The same computation holds when replacing J by −J and choosing σ ∈ Σ − (R 2 ). As the spaces of skew-Killing spinors ϕ corresponding to the standard complex structure J or its opposite on S 2 are each complex 2-dimensional, we deduce that the space of skew Killing spinors with Killing map A + is at least -and therefore exactly -4-dimensional. The same holds for A − . In particular, each skew Killing spinor on S 2 × R 2 is a linear combination with constant coefficients of skew Killing spinors for A + and also one of skew Killing spinors for A − . Note that the vector field ξ, associated to the above-defined skew Killing spinor ψ, is the one coming from the spinor ϕ on S 2 , since T S 2 ≃ Σ + S 2 and Therefore, ξ = ξ S 2 and A 2 ξ = J 2 ξ S 2 = −ξ S 2 , which cannot vanish on the sphere. Thus Aη = 0. If we consider instead of the above constructed ψ the spinor ψ + Y ·ψ for a parallel vector field Y on R 2 with |Y | = 1, we obtain a skew Killing spinor with ξ = −Y , hence Aη = 0.

Classification
Let us first assume that ρ = 1/2 on an open set. By definition of ρ, this condition is equivalent to |ψ + | = |ψ − |. We prove that, under this assumption, the manifold is locally isometric to that in Example 4.1. Proof: Let ψ be a skew Killing spinor of norm one such that |ψ + | = |ψ − |. Then f = 0 by definition of f . Thus η is parallel by Lemma 3.1. In this case η ⊥ is integrable and the spinor ψ restricts to a skew Killing spinor on the integral manifolds. In fact, for any given integral manifold N , its spinor bundle is identified with Σ + M , so the spinor ϕ = ψ + restricts to a skew Killing spinor on N . Indeed, In the next part of the section, we want to exclude the case ρ = 1/2 and make the stronger assumption for any vector field X.
Proof: We take the orthogonal projection of the formulas in Proposition 2.1 to Σ + M and Σ − M . This gives, after using ψ + = ξ · ψ − , dA = 0 and A ∧ AX = 0 that and respectively. Equation (16) gives Hence, by formula (6), we obtain (10). Equation (17) yields Now, by taking the scalar product with ψ − and identifying the real part, the 0-th order term must vanish. This is Equation (12). Also, we have The isomorphism from 2 − M to the orthogonal complement (ψ − ) ⊥ yields Equality (11) from the above identity. Equation (18) gives (13). Finally, Equation (19) yields Taking the Hermitian product with ψ − , we obtain Equations (14) and (15) after identifying the real parts.
In the following, we will further simplify the equations in Lemma 4.4.
for every X ∈ T M .
Remark 4. 6 We can prove integrability conditions analogous to those in Lemma 4.4 and Proposition 4.5 also for arbitrary rank of A. These general conditions are more involved. Since we will not use them in the present paper, we do not state them here. Proof: Assume that Aη = 0 on an open set U . We know that η is a Killing vector field on M . Moreover, by Lemma 3.1, the vector field η has constant length on U . Indeed, for every X ∈ T M , By [6,Thm. 4], since (22) implies Ric(η) = 0, we can conclude that η is parallel on U . But this contradicts item 2 of Lemma 3.1 since f = 0 and A = 0 everywhere by assumption.
In the following, we will often assume assume that Aη = 0 on all of M . If Aη = 0, then we have A 2 η = 0 everywhere, thus the vectors Aη |Aη| and A 2 η |A 2 η| form an orthonormal basis of the image of A. As A is of rank 2, we obtain Furthermore, note that (27) already implies where the last equality comes from the identity (21). Obviously, This equation has been extensively studied in [12]. Using this formula, we now express the Ricci tensor of the vector field Aη.
In the following, we will compute the Ricci curvature of the vector field ( * A)η. Notice first that ( * A)η = η ( * A) = * (η ∧ A). Hence, this vector field belongs to the kernel of A as for any X ∈ T M . Based on the fact Aη ( * A) = * (Aη ∧ A) = 0, we first compute This gives On the other hand, by (26) and (28), we have Comparing the two identities gives the second equation in (30). Equation (31) can be deduced from computing Ric(Aη, ( * A)η)) in two ways from (30) taking the scalar product by ( * A)η in the first formula and by Aη in the second one. Remember that ( * A)η lies in the kernel of A.
In the following, we will establish and prove three technical lemmas (Lemmas 4.9, 4.10 and 4.11), which will show that the kernel and the image of the endomorphism A are integrable and totally geodesic. Then the proof of Theorem A will follow from the de Rham theorem.
Lemma 4.9 Assume that (GA) holds. Then we have the identity Proof: By continuity, it suffices to prove the assertion on the set {p ∈ M | Aη| p = 0} since this set is dense in M by Lemma 4.7. Thus we may assume that Aη = 0 everywhere. For any X ∈ T M , we have where we use Equation (25) in the last equality. Thus, from Lemma 4.8, we find Moreover, δ(A 2 η) = 0. Indeed, for any two-form ω in four dimensions and any vector X, the formula δ(X ω) = * (dX ∧ * ω) − δω(X) holds. Using δA = 0 and 4d(Aη) = ddf = 0, this yields Now, by taking the divergence of both sides of (27), we compute Furthermore, the divergence of Aη is equal to f S/4 as an easy consequence from tracing Equation (25). This finally gives (33).
The following technical lemma expresses a partial trace of the Ricci tensor.
Lemma 4.10 Assume that (GA) holds and that Aη = 0 everywhere. Then the following identity holds: 1 Proof: The proof relies on taking the scalar product of Ric(A 2 η) in Lemma 4.9 with the vector field A 2 η. Indeed, we have Hence, again by (28), we find Finally, the identity Ric(Aη, Aη) which follows from Lemma 4.8, leads to the required equality. Proof: As in the proof of Lemma 4.9, we may assume that Aη = 0 everywhere. By Lemma 4.8 we know that We take the divergence of both sides. We start with the left hand side. Note that for any vector field X ∈ Γ(T M ) the formula δ(Ric(X)) = g(δRic, X) − n i=1 g(Ric(e i ), ∇ ei X) holds, where e 1 , . . . , e n is any pointwise orthonormal basis. Using this and δ(Aη) = f S 4 , we compute To get the divergence of the right hand side, we first compute that of the vector field Ric(( * A)η). For this, we use the same formula as above and again dS = −2δRic to write In the last equality, we used (31). Inserting (26) into (36), we find which in turn gives δ(f · Ric(( * A)η)) = −g(df, Ric(( * A)η)) + f · δ(Ric(( * A)η))) by (30). Comparing Equations (35) and (37), we obtain On the other hand, this sum can be computed on the particular orthonormal frame Aη |Aη| , A 2 η |A 2 η| , e 3 , e 4 with e 3 , e 4 in the kernel of A as follows: using Lemma 4.10, we write Comparing these two computations yields The Cauchy-Schwarz Inequality gives We take the square of this inequality. Then we use (38) and (39) to express the left and the right hand side, respectively. We obtain where besides (21), which says that S = 4|A| 2 , we used |A 2 | 2 = (|A| 2 ) 2 /2, which follows from the fact that A is skew-symmetric of rank two. This inequality is only true if Aη(S) = 0. But then (40) is an equality. Hence, Ric is a multiple of A 2 at every point of   is not constant, then (M, g) is a local Riemannian product S 2 × R 2 around every point and the Killing map equals ±J ⊕ 0.
If, in addition, (M, g) is complete, then (M, g) is globally isometric to the Riemannian product S 2 × Σ 2 , where Σ 2 is either flat R 2 , a flat cylinder with trivial spin structure or a flat 2-torus with trivial spin structure.
Proof: We define U := {p | A p = 0} and U ′ := U ∩ M ′ , U ′′ := U ∩ M ′′ . Recall that U ′ ⊂ U is dense. We know that Equation (29)  But then p ∈ U would imply that S/2 > 0 is an eigenvalue of Ric p and p ∈ W would imply that Ric p = 0, a contradiction.
Note that, as we already noticed in [12, Theorem 2.4], the manifold (M, g) must be globally isometric to the product S 2 × Σ 2 , where Σ 2 is a quotient of flat R 2 . The reason is that the fundamental group of M can act on the S 2 -factor only in a trivial way. It remains to recall that a parallel spinor descends from R 2 to a nontrivial quotient (flat cylinder or torus) if and only if the fundamental group acts on the spin structure of R 2 in a trivial way, that is, the quotient Σ 2 carries the trivial spin structure.

Skew Killing spinors with non-degenerate Killing map A
This section is devoted to the case where we have a skew Killing spinor ψ whose Killing map A is non-degenerate everywhere. Recall that ψ defines a vector field η by (8). As above, we put ρ := |η|.
Here, we want to assume that This is a sensible restriction since M ′′ is dense in M if A is non-degenerate everywhere, see Section 3. Working on M ′′ has the advantage that we do not have to care about the sign of f . Indeed, as explained in Remark 3.3, up to a possible change of orientation on each connected component we may assume that f > 0. In particular, f is defined by ρ = |η| via f = 1 − 4ρ 2 , which will be important for the reverse direction of Proposition 5.1.

Equivalent description by complex structures
Let M be a manifold and A be a skew-symmetric endomorphism field on M . Define a tensor field Proposition 5.1 Let M be a four-dimensional spin manifold and A be a skew-symmetric endomorphism field on M . Put C := C A .
If (M, g) admits a skew Killing spinor ψ associated with A such that M = M ′′ , then there exist an almost Hermitian structure J and a nowhere vanishing vector field η of length |η| =: ρ < 1/2 such that where f := 1 − 4ρ 2 and C P := C(s, Js) for any unit vector s ∈ P , and such that the sectional curvature K P in direction P satisfies where A P := g(As, Js) for any unit vector s ∈ P .
If M is simply-connected, then also the converse statement is true. (41) and (42). Then
In the following computation, the sign '≡' means equality up to a term S(X, Y ) for some symmetric bilinear map S. We compute This implies Using Equations (46) and (47) we obtain (48).
Proof of Prop.5.1: Before we start the proof of the two directions of the assertion, let us first suppose that, on M , we are given a Hermitian structure J and a nowhere vanishing vector field η of length ρ < 1/2. We want to define a vector field ξ such that the identities ξ = −(|ξ|/ρ) · η and ρ = |ξ|/(1 + |ξ| 2 ) hold according to Equation (9). Since this leads to a quadratic equation, we have to choose one of the solutions. Here we use our assumption M = M ′′ and define f = 1 − 4ρ 2 and ξ = 2(f − 1) −1 η, compare Remark 3.3, which motivates this choice. Assume that the orientation on M is such that orthonormal bases of the form s 1 , Js 1 , s 2 , Js 2 are negatively oriented. We define a one-dimensional subbundle E of ΣM by We want to show that E is parallel with respect to∇ defined by∇ X ϕ := ∇ X ϕ − AX · ϕ if and only if J and η satisfy (41) and (42). Let X and Y be vector fields satisfying ∇X = ∇Y = 0 at p ∈ M . Then we have at p ∈ M This equals iX · (∇ Y ϕ) − if and only if (∇ Y J)(X) = 2X Jξ ∧ AY + ξ ∧ JAY holds, which is equivalent to Equation (41). Furthermore, This equals ξ · (∇ X ϕ) − if and only if ∇ X ξ = (1 − |ξ| 2 )AX − 2g(Aξ, X)ξ holds, which is equivalent to (42). Consequently, E is parallel with respect to∇ if and only if J and η satisfy (41) and (42).
Assume that∇ reduces to a connection∇ E on E. Then Equations (41) and (42), and therefore also (46), (47) and (48) hold. We will show that the curvatureR of∇ E vanishes if and only if the Riemannian curvature R of M equals the tensor B defined by for all vector fields X and Y on M . By an easy calculation similar to that in the proof of Proposition 2.1, we getR

This shows thatR vanishes if and only if
for all vector fields X and Y and all sections ϕ of E. In the following, we will use that 2 ± M acts trivially on Σ ∓ M and that, for any nowhere vanishing section ϕ ± of Σ ± M , the maps defined by (7) are isomorphisms. Let ϕ be a section of E such that ϕ + (x) = 0, ϕ − (x) = 0 for all x ∈ M (here we use that ξ does not vanish). Then Thus (50) for all X, Y ∈ X(M ) and all Z ∈ Γ(P ). Recall that (46) holds in our situation, which we will use in the following computations. Equations (55) and (56) are equivalent to the two equations which are equivalent to (43) and (44), respectively. Because of Now we can prove both directions of the proposition. Suppose that there exists a spinor field ψ on M satisfying ∇ X ψ = AX · ψ for all X ∈ T M such that M = M ′′ . The latter condition means that the vector field η defined in (8) satisfies 0 < ρ = |η| < 1/2. In particular, ψ − = 0 everywhere and we can define an almost Hermitian structure J by J(X) · ψ − = iX · ψ − . Thus we may apply our above considerations. If we define E ⊂ ΣM and∇ as above, then ψ is a∇-parallel section of E. In particular,∇ reduces to a connection∇ E and the curvature of∇ E vanishes thus (41) -(45) hold.
Conversely, if we are given an almost Hermitian structure J and a nowhere vanishing vector field η of length 0 < ρ = |η| < 1/2 such that (41) -(45) are satisfied. Then we can define a onedimensional subbundle E ⊂ ΣM by (49) together with a flat covariant derivative∇ on E. If M is simply-connected, then E admits a parallel section, which is a skew Killing spinor.

Remark 5.3
Let J be an almost Hermitian structure on a four-dimensional manifold M such that (41) and (42) hold for a skew-symmetric endomorphism field A and a vector field E. Then J defines a reduction of the SO(4)-bundle SO(M ) to U(2). Here we want to give the intrinsic torsion of this bundle in the special case where A and J commute. The two components of the intrinsic torsion of this bundle are the Nijenhuis tensor N of J and the differential dΩ of the Kähler form Ω := g(J·, ·). A direct calculation using (41) and (42) shows that under the assumption AJ = JA these components are given by N = 0 and dΩ = −2A ∧ (ξ Ω).

The case where Aη is parallel to Jη
Let us assume again that the Killing map A is non-degenerate everywhere. We want to consider the case where Aη is parallel to Jη in more detail. We will see that, in this situation, the existence of skew Killing spinors is related to doubly warped products and to local DWP-structures. These notions and their basic properties are explained in the appendix.

Lemma 5.4
Assume that M admits a skew Killing spinor with nowhere vanishing Killing map A that satisfies Aη = uJη for some function u. Then A 2 η = −u 2 η. In particular, AJ = JA.
Proof. Note first that Lemma 3.1, 4 and Eq. (46) give for all X, Y ∈ T M . Consequently, f ∇ η A = 0. Because of Eq. (41) gives (∇ η J)η = 0. Now, by differentiating the equality Aη = uJη in the direction of η, we get Finally, using the fact that ∇ η η = f Aη and f ∇ η A = 0, we get that η(u) = 0 and f 2 A 2 η = −u 2 f 2 η. The latter equation implies Let (M 3 ,ĝ,η) be a minimal Riemannian flow, i.e., an orientable three-dimensional Riemannian manifold together with a unit Killing vector fieldη. Then, locally, (M ,ĝ) is a Riemannian submersion over a two-dimensional base manifold B. Let us fix a Hermitian structureĴ onη ⊥ and put ω :=ĝ(·,Ĵ·). We define a functionτ onM which is constant along the fibres by∇ Xη =τ ·Ĵ(X) for X ∈η ⊥ . Furthermore, letK denote the Gaussian curvature of B. Now consider the metric g rs = r 2ĝη ⊕ s 2ĝη ⊥ onM , whereĝη,ĝη⊥ are the components of the metricĝ along Rη andη ⊥ , respectively. Then (M , g rs , r −1η ) is again a minimal Riemannian flow and we obtain new functionŝ τ andK, sayτ rs andK rs . These functions satisfŷ If our four-dimensional manifold M is endowed with a DWP-structure, then every three-dimensional leaf associated with this structure can be understood as a minimal Riemannian flow. In this way, we obtain functions τ and K on M .
Conversely, suppose that M is simply-connected and admits a local DWP-structure (ν, η) on M such that the length ρ of η satisfies 0 < ρ < 1/2. Moreover, assume that K and τ satisfy (58) for f := 1 − 4ρ 2 . Then M admits a skew Killing spinor such that η is associated with ψ according to (8) and such that Aη||Jη.
Proof: Assume first that M admits a skew Killing spinor such that Aη||Jη and 0 < ρ < 1/2 everywhere. We define a vector field ν and functions A E and A P by Then η is a Killing vector field, see Remark 3.2. Equation (42) yields We want to show that (ν, η) is a DWP-structure. The next Lemma will prove all properties of such a structure except the conditions for the Weingarten map W = −∇ν and its eigenvalues.
Indeed, (46) implies g C(s 1 , s 2 ), η = 0, thus we obtain which gives s 1 (A E ) = 0. Using (65) and taking into account that [s 1 , s 2 ] is a multiple of s 1 , we get Hence we proved that besides ρ also A E and A P are constant on the integral manifolds of ν ⊥ . Thus also µ and λ are constant along these leaves. Consequently, (ν, η) is a local DWP-structure on M . By (62), the associated function τ satisfies where s ∈ {η, ν} ⊥ is of length one. This proves the first equation in (58).
It remains to prove that also the second equation in (58) is true. Let N be an integral manifold of ν ⊥ . Then, locally, N is a Riemannian submersion over a base manifold B. The following lemma will relate the sectional curvature K P in direction of P = span{s 3 , s 4 } to the Gaussian curvature K of B, which will almost finish the proof of the forward direction of Theorem 5.5.
Lemma 5.7 Let (ν, η) be a local DWP-structure such that the coefficients of the Levi-Civita connection satisfy (64) with respect to an orthonormal frame s 1 = −η/ρ, s 2 = ν, s 3 , s 4 . Then the Gaussian curvature K of B equals Proof Let A denote the fundamental tensor used in O'Neill's formulas. We have The O'Neill formula for R N now gives which combined with (68) implies the assertion.
Up to rescaling of the metric, each integral manifold N in our construction has a Sasakian structure, see [7] for a definition of such structures. Indeed, η restricted to N is a Killing vector field of constant length and ∇η restricted to η ⊥ equals |η|τ J| η ⊥ , where also τ is constant. The Nijenhuis tensor of J| η ⊥ vanishes since η ⊥ is two-dimensional. Consequently,ξ := η/(τ |η|) is the Reeb vector field of a Sasakian structure on (N,g := τ 2 g). The scalar curvature of (N,g) equals S = 4λ/(f τ )+2.
Thus we are up to a change of orientation exactly in the situation described above.
In dimension three Sasakian quasi-Killing spinors of type can also be understood as transversal Killing spinors, see [11] for a definition. If we return to our original metric g on N , this means that the restrictions of ψ ± to N are transversal Killing spinors. Indeed, holds for the transversal covariant derivative∇ on N .
It remains to notice thatη must be a Killing vector field along (M ,ĝ) since it is already Killing on (M, g) and is tangent toM . On the whole, we obtain the doubly warped product metric as required.