A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

It is a classical fact that the cotangent bundle T∗M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^* {\mathcal {M}}$$\end{document} of a differentiable manifold M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}$$\end{document} enjoys a canonical symplectic form Ω∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^*$$\end{document}. If (M,J,g,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathcal {M}},\mathrm{J} ,g,\omega )$$\end{document} is a pseudo-Kähler or para-Kähler 2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2n$$\end{document}-dimensional manifold, we prove that the tangent bundle TM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T{\mathcal {M}}$$\end{document} also enjoys a natural pseudo-Kähler or para-Kähler structure (J~,g~,Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\tilde{\hbox {J}}},\tilde{g},\Omega )$$\end{document}, where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is the pull-back by g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document} of Ω∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^*$$\end{document} and g~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} is a pseudo-Riemannian metric with neutral signature (2n,2n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2n,2n)$$\end{document}. We investigate the curvature properties of the pair (J~,g~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\tilde{\hbox {J}}},\tilde{g})$$\end{document} and prove that: g~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} is scalar-flat, is not Einstein unless g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document} is flat, has nonpositive (resp. nonnegative) Ricci curvature if and only if g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document} has nonpositive (resp. nonnegative) Ricci curvature as well, and is locally conformally flat if and only if n=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1$$\end{document} and g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document} has constant curvature, or n>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>2$$\end{document} and g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document} is flat. We also check that (i) the holomorphic sectional curvature of (J~,g~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\tilde{\hbox {J}}},\tilde{g})$$\end{document} is not constant unless g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document} is flat, and (ii) in n=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1$$\end{document} case, that g~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{g}$$\end{document} is never anti-self-dual, unless conformally flat.


Introduction
sponding structure on T M can also be constructed using the splitting H M ⊕ V M induced by the Levi-Civita connection of the Kählerian metric.
However, it turns out that * is not compatible withJ * , since it turns out that * (J * .,J * .) = −ε * instead of the required formula * (J * .,J * .) = ε * (here and in the following, in order to deal simultaneously with the complex and para-complex cases, we define ε to be such that (J * ) 2 = −εId, i.e. ε = 1 in the complex case and ε = −1 in the para-complex case). It follows that the tensorg * is not symmetric and therefore we failed in constructing a canonical pseudo-Riemannian structure on T * M.
On the other hand, the same idea works well if one considers, instead of the cotangent bundle, the tangent bundle of a pseudo-or para-Kähler manifold (M, J, g), thus obtaining a canonical pseudo-or para-Kähler structure. The purpose of this note is to investigate in detail this construction and to study its curvature properties. The results are summarized in the following: -The pseudo-Riemannian metricg has the following properties: Remark 1 We use in (iv) the general property that four-dimensional neutral pseudo-Kähler or para-Kähler manifolds are self-dual if and only if their scalar curvature vanishes. This is analogous to the case of Kähler four-dimensional manifolds, except that self-duality is exchanged with anti-self-duality. A proof of this statement is given in Theorem 7.2 in the appendix.
This result is a generalization of previous work on the tangent bundle of a Riemannian surface (see [2,9,10]). The authors wish to thank Brendan Guilfoyle for his valuable suggestions and comments.

Almost complex and para-complex structures on the tangent bundle
Given a manifold M endowed with an almost complex or almost para-complex structure J, it is only natural to ask whether its tangent or cotangent bundle inherit such a structure. The answer is positive: be an almost complex (resp. para-complex) manifold. Then its tangent bundle admits a canonical almost complex (resp. para-complex) structurẽ J. Furthermore, if J is complex (resp. para-complex), so isJ.
Remark 2 Such a result has been proven already by Lempert and Szöke [13] for the tangent bundle in the almost complex case. Their construction uses the jets over M and is quite a bit more technical than our proof. However it gives an interesting interpretation of the meaning ofJ. We shall see below in Proposition 2 a different and simpler way of defining and understandingJ, provided M is a pseudo-or para-Kähler manifold.
Proof We prove the result using coordinate charts, which amounts to showing thatJ can be defined independently of any change of variable. Let y = ϕ(x) be a local change of coordinates on R n and write ξ and η respectively for the tangent coordinates induced by the charts (i.e. i ξ i ∂/∂ x i = i η i ∂/∂y i ). The change of tangent coordinates at x is ξ → η = dϕ(x)ξ , in other words ϕ induces a chart on R 2n , : (x, ξ) → (ϕ(x), dϕ(x)ξ ). The tangent coordinates at (x, ξ) (resp. (y, η)) are denoted by (X, ) (resp. (Y, H )) and the change of (doubly) tangent coordinates is Assume moreover that we have a (1, 1) tensor, which reads in the x coordinate as the matrix J(x) and in the y coordinate as the matrix J (y) where (D ξ J)(x) denotes in this proof the directional derivative of the matrix J at x in the direction ξ (not a covariant derivative).
We now define the (1, 1) tensorJ in the (x, ξ) coordinate bỹ Let us prove that this definition is coordinate-independent (for greater readability we will often write J, J for J(x), J (y)). Using (1) and the symmetry of the second order differential d 2 ϕ(x), whereJ denotes the map corresponding toJ in the (y, η) coordinates. Consequently the tensor on M extends naturally to T M. We have so far defined a (1, 1) tensor on T M without extra assumptions. Suppose now that J is an almost complex (resp. para-complex) structure, so that J 2 = −εId. Differentiating this property yields J D ξ J + (D ξ J) J = 0. Theñ so thatJ is also an almost complex (resp. para-complex) structure. Finally if J is a complex (resp. para-complex) structure then we can use complex (resp. para-complex) coordinate charts, which amounts to saying that J is a constant matrix. ThenJ defined in the associated charts on T M takes a simpler expression, and is also constant:J and that characterizes a complex (resp. para-complex) structure.

Remark 3
Finding a similar almost-complex structure on T * M is much more difficult, and may not be true in all generality. The Reader will note that, whenever M is endowed with a pseudo-Riemannian metric, we have a musical correspondence between T M and T * M, andJ induces a corresponding structureJ * on T * M. However different metrics will yield different structures on T * M. There is one unambiguous case, which will be the setting in the remainder of this article, namely when J is integrable.

The Kähler structure
Let M be a differentiable manifold. We denote by π and π * the canonical projections T M → M and T M * → M. The subbundle ker(dπ) := V M of T T M (it is thus a bundle over T M) will be called the vertical bundle. Assume now that M is equipped with a linear connection ∇. The corresponding horizontal bundle is defined as follows: letX be a tangent vector to T M at some point (x 0 , V 0 ). This implies that there exists a curve γ (s) , we define the connection map (see [2,6] where ∇ denotes the Levi-Civita connection of the metric g. If X is vertical, we may assume that the curve γ stays in a fiber so that V (s) is a curve in a vector space. We then define KX to be simply V (0). The horizontal bundle is then K er(K) and we have a direct sum Here and in the following, is a shorthand notation for dπ .

Lemma 1 [6]
Given a vector field X on (M, ∇) there exists exactly one vector field X h and one vector field where R denotes the curvature of ∇ and we use the direct sum notation (2).
The Reader should not confuse the horizontal lift X h , which is a vector field on T M constructed from a vector field X ∈ X(M), with the notationX h = X denoting the horizontal part ofX ∈ X(T M). Similarly, the vertical lift X v is not the vertical We say that a vector fieldX on T M is projectable if it is constant on the fibres, i.e.
According to the lemma above, it is equivalent to the fact that there exists two vector fields X 1 and X 2 on M such thatX = ( Assume now that M is equipped with a pseudo-Riemannian metric g, i.e. a nondegenerate bilinear form. By the non-degeneracy assumption, we can identify T * M with T M by the following (musical) isomorphism: where ξ is defined by The Liouville form α ∈ 1 (T * M) is the 1-form defined by α (x,ξ ) (X ) = ξ x (dπ * (X )), whereX is a tangent vector at the point (x, ξ) of T * M. The canonical symplectic form on T M * is defined to be * := −dα. There is an elegant, explicit formula for the symplectic form := ι * ( * ) in terms of the metric g and the splitting induced by the Levi-Civita connection ∇ (see [1,12]):

Lemma 2 LetX andȲ be two tangent vectors to T M; we have
. .

Moreover,g is symmetric and therefore defines a pseudo-Riemannian metric on T M.
Proof of Proposition 2 Let us write the splitting of T T M in a local coordinate x as in the proof of Proposition 1 ( 3 ). The Levi-Civita connection is expressed through its connection form μ: Because J is integrable, we may choose x to be a complex coordinate, so that J is a constant endomorphism, and dJ(x)ξ vanishes. Because M is Kähler, we know that μ(X ) commutes with J. However, ∇ being without torsion,

Corollary 2
The symplectic form is compatible with the complex or para-complex structureJ.
Proof Using Lemma 2, we compute

The Levi-Civita connection ofg
The following lemma describes the Levi-Civita connection∇ ofg in terms of the direct decomposition of T T M, the Levi-Civita connection ∇ of g and its curvature tensor R.

Lemma 3 LetX andȲ be two vector fields on T M and assume thatȲ is projectable, then at the point
where Moreover, if M is a pseudo-Riemannian surface with Gaussian curvature c, we have Proof We use Lemma 1 together with the Koszul formula: Moreover, taking into account thatg(Y v , Z h ) and similar quantities are constant on the fibres, we obtain Finally, introducing From (3) and (4) we deduce the required formula for∇XȲ .
, hence the tensor T 1 becomes: Remark 4 It should be noted that covariant derivatives with respect to a projectable vertical field X v always vanish.

Proposition 3 The structureJ is parallel with respect to∇.
Proof It can be seen as a trivial consequence of the fact thatJ is complex (resp. paracomplex) and is closed, but can also be checked directly, using the equivariance properties of J w.r.t. the connection ∇ and the curvature tensor R. Using the definition ofJ and Lemma 3,∇XJȲ is obvious provided where we have used Bianchi's identity.

Proposition 4
The curvature tensor Rm := −g(R., .) ofg at (x, V ) is given by the formula Proof of Proposition 4 We will compute the curvature tensor for projectable vector fields, and need only do so for the following six cases, due to the symmetries of Rm. Remark 4 simplifies computations greatly, since most vertical derivatives vanish, except when the derived vector field is not projectable. In particularR(X v , Y v ) vanishes as endomorphism, hence:

Moreover, (T M,g) is scalar flat and the Ricci tensor ofg is
To obtain the last three combinations, let us first deriveR(X h , Y h )Z h . This is more delicate since we have to covariantly differentiate non-projectable quantities. Indeed Recalling the lemma 4 in [11], there exists a vector field U on M such that U (x) = V and (∇ X U )(x) = 0. Then the vertical lift of T 1 (Y, Z , U ) is seen to agree to first order with thus allowing us to use the formula in Lemma 3: Summing up, From that we deduce directly On the other hand, using repeatedly Remark 4, The claimed formula is easily deduced using the symmetries of the curvature tensor. In order to calculate the Ricci curvature ofg, we consider a Hermitian pseudoorthonormal basis (e 1 , . . . , e 2n ) of T x M, i.e. g(e a , e b ) = ε a δ ab , where ε a = ±1, and e n+a = Je a . In particular, ε n+a = εε a .This gives a (non-orthonormal) basis of T (x,V ) T M:ē a := (e a ) hē2n+a := (e a ) v .
A calculation using Corollary 1 shows that the expression ofg in this basis is: (ε n+1 , . . . , ε 2n ). It follows that Ric(X v , Y v ) and Moreover, noting thatg μν =g μν , We see easily that Ric vanishes whenever one of the vectors is along the vertical fiber, thus the expected formula.

The Weyl curvature tensor ofg
Proposition 5 The Weyl tensor W at (x, V ) is given by W )).
In particular, if n = 1, Remark 5 This result has been proved in the case n = 1 and ε = 1 in [9].
Proof of Proposition 5 Since the scalar curvature vanishes, we have where denotes the Kulkarni-Nomizu product. Recall that Ric(X ,Ȳ ) = 0 if one of the two vectorsX andȲ is vertical. Consequently The expression of the Weyl tensor follows easily. In the case n = 1 of a surface with Gaussian curvature c, we have Ric(X, Y ) = cg(X, Y ) and Rm(X, Y, Z , W ) = c(g(X, Z )g(Y, W ) − g(X, W )g(Y, Z )). Hence using Proposition 4, the expression of Weyl tensor simplifies and we get the claimed formula.

Proof of Corollary 4
We first deal with the case n = 1. Lemma 3 implies that which vanishes if and only if (X.c)Y = (Y.c)X for all vectors X, Y , i.e. the curvature c is constant. Analogously, if ε = −1, which again vanishes if and only if the curvature c is constant. Assume now that (T M,g) is conformally flat with n ≥ 2. Thus in particular vanishes, so JW )).
(Observe that this equation always holds if M is a surface.) Let us apply the symmetry property of the curvature tensor to this equation with Z = X and JW = Y , assuming furthermore that X and Y are two non-null vectors: The set of non null vectors being dense in T M, it follows by continuity that g is Einstein. We deduce that so g has constant curvature. But since M is Kähler and has dimension 2n ≥ 4, it must be flat.
Finally, we recall the general result linking the Weyl tensor to the scalar curvature in dimension four: for a neutral pseudo-Kähler or para-Kähler metric, self-duality is equivalent to scalar flatness (see Theorem 7.2 in annex). We can therefore conclude

Corollary 5 In dimension four (n = 1), the metricg is anti-self-dual if and only the curvature c of g is constant.
Proof Thanks to Proposition 4, we know thatg is scalar flat, hence self-dual (W − vanishes identically). In order forg to be also anti-self-dual, the Weyl tensor has to vanish completely, which amounts, following Corollary 4, to having constant (sectional) curvature c on M.

5.3
The holomorphic sectional curvature of (J,g)

Proposition 6 (J,g) has constant holomorphic sectional curvature if and only if g is flat.
Proof Define the holomorphic sectional curvature tensor ofg by Hol(X ) := Rm(X ,JX ,X ,JX ). Writing any doubly tangent vectorX as the sum of a horizontal and a vertical factor, we will compute Hol(X h + Y v ). We deduce from Proposition 4 that Rm vanishes whenever two or more entries are vertical. Hence, using the antisymmetric properties of the Riemann tensor w.r.t. the complex or para-complex structure, In particular, It follows from the first equation that if Hol is constant, it must be zero. Hence, from the second and third equation we deduce that Hol must vanish, i.e. g is flat.

Examples
The simplest examples where we may apply the construction above is where (M, J, g, ω) is the plane R 2 equipped with the flat metric g := dq 2 1 + εdq 2 2 and the complex or para-complex structure J defined by J(∂ q 1 , ∂ q 2 ) = (−ε∂ q 2 , ∂ q 1 ). In other words, R 2 is identified with the complex plane C or the para-complex plane D. We recall that D, called the algebra of double numbers, is the two-dimensional real vector space R 2 endowed with the commutative algebra structure whose product rule is given by The number (0, 1), whose square is (1, 0) and not (−1, 0), will be denoted by τ .
We claim that in the complex case ε = 1, the structure (J,g, ) just constructed on T C is equivalent to that of the standard complex pseudo-Euclidean plane (C 2 ,J, ., . 2 , ω 1 ), whereJ is the canonical complex structure, (z 1 = x 1 + iy 1 , z 2 = x 2 + iy 2 ) are the canonical coordinates and ., . 2 To see this, it is sufficient to consider the following complex change of coordinates which preserves the symplectic form, since we have where is the canonical symplectic form of T * C g T C. The metric of a pseudo-Kähler structure being determined by the complex structure and the symplectic form through the formulag = (.,J.), we have the required identification. Analogously, in the para-complex case ε = −1, the structure (J,g, ) constructed on T D is equivalent to that of the standard para-complex plane (D 2 ,J, ., . * , ω * ), whereJ is the canonical para-complex structure, (w 1 = u 1 + τ u 1 , w 2 = u 2 + τ y 2 ) are the canonical coordinates and ., .
Here we have to be careful with the identification of T * D with T D: since the metric g is dq 2 1 − dq 2 2 , we have q 1 := dp 1 g ∂ p 1 and q 2 := dp 2 g −∂ q 2 . Hence * = dq 1 ∧ dp 1 + dq 2 ∧ dp 2 and = dq 1 ∧ dp 1 − dq 2 ∧ dp 2 . Introducing the change of para-complex coordinates we check that Their tangent bundles enjoy a interesting geometric interpretation (see [9]): the tangent bundle T S 2 is canonically identified with the set of oriented lines of Euclidean three-space: Analogously, the tangent bundle T H 2 is canonically identified with the set of oriented negative (timelike) lines of three-space endowed with the metric ., . 1 := dx 2 1 + dx 2 2 − dx 2 3 : Finally, the tangent bundle T dS 2 is canonically identified with the set of oriented positive (spacelike) lines of three-space endowed with the metric ., . 1 : Observe that the metric constructed on T S 2 (resp. T H 2 ) has non-negative (resp. nonpositive) Ricci curvature.

Appendix: the Weyl tensor in the pseudo-Kähler or para-Kähler cases
The Riemann curvature tensor Rm of a pseudo-Riemannian manifold N may be seen as a symmetric form R on bivectors of 2 T N (see [3] for references). Splitting R along the eigenspaces + ⊕ − of the Hodge operator * on 2 T N , yields the following block decomposition where Z * denotes the adjoint w.r.t. the induced metric on 2 T N , so that W = W + ⊕ W − , the Weyl tensor seen as a 2-form on 2 T N , is the traceless, Hodge-commuting part of the Riemann curvature operator R. Hence the following formula If, additionally, N is a four dimensional Kähler manifold, then The result for neutral pseudo-Kähler manifolds is probably known and relates to representation theory (see [3] for introduction and references), but since we could not find an explicit proof in the literature 5 , we will give a simple one below. To our knowledge, the proof for the para-Kähler case is new (albeit similar).

A.1 The pseudo-Kähler case
We will write explicitly the Weyl tensor in a given positively oriented orthonormal frame, denoted by (e 1 , e 1 , e 2 , e 2 ), where e 1 = Je 1 , e 2 = Je 2 , g(e 1 ) = g(e 1 ) = −1 and g(e 2 ) = g(e 2 ) = +1. (For brevity, g(X ) denotes the norm g(X, X ).) The pseudometric g extends to bivectors, has signature (2, 4), and will be again denoted by g: g(e a ∧e b ) = g(e a )g(e b )−g(e a , e b ) 2 = g(e a )g(e b ), so that B = (e 1 ∧e 1 , e 1 ∧e 2 , e 1 ∧ e 2 , e 1 ∧ e 2 , e 1 ∧ e 2 , e 2 ∧ e 2 ) is an orthonormal frame of 2 , with g(e a ∧ e b ) = −1, except for g(e 1 ∧ e 1 ) = g(e 2 ∧ e 2 ) = +1. (Note that the other convention, taking −g does not change the induced metric on 2 .) Since the volume e 1 ∧e 1 ∧e 2 ∧e 2 is positively oriented, we construct an orthonormal eigenbasis for the Hodge star on 2 T N :  where e ab stands for e a ∧ e b , for greater legibility. We have written the matrix as a table for clarity and to make symmetries more obvious, and because R is symmetric we need only write half the matrix. We have used the internal symmetries of R, to choose among equivalent coefficients the ones lowest in the lexicographic order of the indices. The Weyl tensor satisfies some of the J-symmetries of R: indeed Hence the following symmetries (fewer than for Rm) in the coefficients of Ric g, g g and Rm, and therefore W: Expanding on the above eigenbasis of + ⊕ − (which differs from the one in the positive definite case) yields the following Weyl tensor coefficients, which we have simplified using the symmetries above (up to a factor 1/2 due to normalization):  We let the Reader check that in the neutral case, the W + part is not a multiple of the scalar curvature, which completes the proof of Theorem 7.2.

A.2 The para-Kähler case
The computations are almost identical, but the results differ from the pseudo-Kähler setup, because the para-complex structure J is now an anti-isometry: R(JX, JY )Z = −R(X, Y )Z . We pick an orthonormal basis (e 1 , e 1 , e 2 , e 2 ) with e 1 = Je 1 , e 2 = Je 2 , and g(e 1 ) = g(e 2 ) = +1, g(e 1 ) = g(e 2 ) = −1. The frame B = (e 1 ∧e 1 , e 1 ∧e 2 , e 1 ∧ e 2 , e 1 ∧ e 2 , e 1 ∧ e 2 , e 2 ∧ e 2 ) of 2 T N is also orthonormal w.r.t. the induced metric on 2 , again denoted by g, which has signature (2, 4): g(e a ∧e b ) = g(e a )g(e b ) = −1, except for g(e 1 ∧ e 2 ) = g(e 1 ∧ e 2 ) = +1. An orthonormal eigenbasis for the Hodge operator is the following: where the E + a (resp. E − a ) span + (resp. − ). (Note the sign differences w.r.t. the pseudo-Kähler case.) Since J is anti-isometric and parallel, Hence the following symmetries of the Riemannian curvature operator R, expressed in the frame B (for symmetry reasons and greater legibility, lower left coefficients are not written in this and the subsequent matrices):  and the same property holds for g g. Hence the following symmetries (fewer than for Rm) in the coefficients of Ric g, g g and Rm, and therefore W: Let us now express W in the Hodge basis defined earlier, using the above symmetries (up to a factor 1/2 due to normalization).