A canonical structure on the tangent bundle of a pseudo- or para-K\"ahler manifold

It is a classical fact that the cotangent bundle $T^* \M$ of a differentiable manifold $\M$ enjoys a canonical symplectic form $\Omega^*$. If $(\M,\j,g,\omega)$ is a pseudo-K\"ahler or para-K\"ahler $2n$-dimensional manifold, we prove that the tangent bundle $T\M$ also enjoys a natural pseudo-K\"ahler or para-K\"ahler structure $(\J,\G,\Omega)$, where $\Omega$ is the pull-back by $g$ of $\Omega^*$ and $\G$ is a pseudo-Riemannian metric with neutral signature $(2n,2n)$. We investigate the curvature properties of the pair $(\J,\G)$ and prove that: $\G$ is scalar-flat, is not Einstein unless $g$ is flat, has nonpositive (resp.\ nonnegative) Ricci curvature if and only if $g$ has nonpositive (resp.\ nonnegative) Ricci curvature as well, and is locally conformally flat if and only if $n=1$ and $g$ has constant curvature, or $n>2$ and $g$ is flat. We also check that (i) the holomorphic sectional curvature of $(\J,\G)$ is not constant unless $g$ is flat, and (ii) in $n=1$ case, that $\G$ is never anti-self-dual, unless conformally flat.


Introduction
It is a classical fact that given any differentiable manifold M, its cotangent bundle T * M enjoys a canonical symplectic structure Ω * .
Moreover, given a linear connection ∇ on a manifold M, (e.g. the Levi-Civita connection of a Riemannian metric), the bundle T T M splits into a direct sum of two subbundles HM and V M, both isomorphic to T M. This allows to define an almost complex structure J by setting J(X h , X v ) := (−X v , X h ), where, for X ∈ T T M = HM ⊕ V M, we write X ≃ (X h , X v ) ∈ T M × T M. Analogously, one may introduce a natural almost para-complex (or bi-Lagrangian) structure, setting J ′ (X h , X v ) := (X v , X h ). 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: Main Theorem Let (M, J, g, ω) be a pseudo-or para-Kähler manifold. Then T M enjoys a natural pseudo-or para-Kähler structure (J,g, Ω) with the following properties: -J is the canonical complex or para-complex structure of T M induced from that of M; -Ω is the pull-back of Ω * by the metric isomorphism T M ≃ g T * M; -The pseudo-Riemannian metricg can be recovered fromJ and Ω by the equationg(., .) := Ω(.,J.); -According to the splitting T T M = HM ⊕ V M induced by the Levi-Civita connection of g, the triple (J,g, Ω) takes the following expression: -The pseudo-Riemannian metricg has the following properties: (i)g has neutral signature neutral (2n, 2n) and is scalar flat; (ii) (T M,g) is Einstein if and only if (M, g) is flat, and therefore (T M,g) is flat as well; (iii) the Ricci curvature Ric ofg has the same sign as the Ricci curvature Ric of g; (iv) (T M,g) is locally conformally flat if and only if n = 1 and g has constant curvature, or n > 2 and g is flat; if n = 1,g is always self-dual, so anti-self-duality is equivalent to conformal flatness; (v) the pair (J,g) has constant holomorphic curvature if and only if g is flat.
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 A.2 in the appendix.
This result is a generalization of previous work on the tangent bundle of a Riemannian surface (see [10], [11], [3]). The authors wish to thank Brendan Guilfoyle for his valuable suggestions and comments.
1 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. paracomplex) structureJ. Furthermore, if J is complex (resp. para-complex), so isJ.
Remark 2. Such a result has been proven already by Lempert & Szöke [14] 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 Φ on R 2n , Φ : (x, ξ) → (ϕ(x), dϕ(x)ξ). The tangent coordinates at (x, ξ) (resp. (y, η)) are denoted by (X, Ξ) (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 Differentiating this equality along ξ yields 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.
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: 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 structurẽ J * 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) = (x(s), V (s)) such that (x(0), V (0)) = (x 0 , V 0 ) and γ ′ (0) =X. If X / ∈ V M (which implies x ′ (0) = 0), we define the connection map (see [7], 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 Ker(K) and we have a direct sum Here and in the following, Π is a shorthand notation for dπ.
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 = ΠX denoting the horizontal part ofX ∈ X(T M). Similarly, the vertical lift X v is not the vertical projectionX v = KX.
We say that a vector fieldX on T M is projectable if it is constant on the fibres, i.e. (ΠX, KX)(x, V ) = (ΠX, KX)(x, V ′ ). 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 non-degenerate 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 ∈ Ω 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 Ω * := −dα. There is an elegant, explicit formula for the symplectic form Ω := ι * (Ω * ) in terms of the metric g and the splitting induced by the Levi-Civita connection ∇ (see [2], [13]):   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, µ(X)Y = µ(Y )X, so K(J(X, Ξ)) = JΞ + Jµ(X)ξ = JK(X, Ξ).
Corollary 2. The symplectic form Ω is compatible with the complex or paracomplex structureJ.
Proof. Using Lemma 2, we compute The Reader should be aware of the conflicting notation: the splitting of T T M ≃ R 4n as R 2n ⊕ R 2n induced by the coordinate charts (e.g.X ≃ ((x, ξ), (X, Ξ))) differs a priori from the connection-induced splittingX ≃ (ΠX, KX).

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 (x, V ) we have Proof. We use Lemma 1 together with the Koszul formula: where X, Y and Z are three vector fields on T M. From the fact that Moreover, taking into account thatg(Y v , Z h ) and similar quantities are constant on the fibres, we obtain Finally, introducing from which we deduce that 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. para-complex) and Ω 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. where
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: 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 [12], 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 pseudo-orthonormal 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 (nonorthonormal) basis of T (x,V ) T M: e a := (e a ) hē2n+a := (e a ) v .
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.
Assume now that (T M,g) is conformally flat with n ≥ 2. Thus in particular (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 A.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.

4.3
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, Hol(X h + (JX) v ) = g(T 2 (X, JX, X, V ), JX) + 4εHol(X).
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(∂ q1 , ∂ q2 ) = (−ε∂ q2 , ∂ q1 ). 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 twodimensional 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 .
To see this, it is sufficient to consider the following complex change of coordinates ), which preserves the symplectic form, since we have where Ω 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 = Ω(.,J.), we have the required identification.
The next simplest examples of pseudo-Riemannian surfaces are the twodimensional space forms, namely the sphere S 2 , the hyperbolic plane H 2 := {x 2 1 + x 2 2 − x 2 3 = −1} and the de Sitter surface dS 2 := {x 2 1 + x 2 2 − x 2 3 = 1}. Their tangent bundles enjoy a interesting geometric interpretation (see [10]): 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. non-positive) Ricci curvature.

A 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 Λ 2 T N (see [4] for references). Splitting R along the eigenspaces Λ + ⊕ Λ − of the Hodge operator * on Λ 2 T N , yields the following block decomposition where Z * denotes the adjoint w.r.t. the induced metric on Λ 2 T N , so that W = W + ⊕ W − , the Weyl tensor seen as a 2-form on Λ 2 T N , is the traceless, Hodgecommuting part of the Riemann curvature operator R. Hence the following formula W = Rm − 1 2 Ric g + Scal 12 g g .

If, additionally, N is a four dimensional Kähler manifold, then
Theorem A.1. W + can be written as a multiple of the scalar curvature by a parallel non-trivial 2-form on Λ 2 T N .
See Prop. 2 in [6] for a proof and the explicit formula for the tensor involved. We do not need it explicitly since we are only interested in the following Corollary 6. (N , g, J) is anti-self-dual (W + = 0) if and only if the scalar curvature vanishes.
The result extends to the two cases considered in this article: (1) neutral pseudo-Kähler manifolds and (2) para-Kähler manifolds, with a slight twist: W + is replaced by W − . Precisely: Let (N , g, J) be a four dimensional manifold endowed with a pseudo-Kähler neutral metric (respectively a para-Kähler metric, necessarily neutral). Then the Weyl tensor W commutes with the Hodge operator and N is self-dual (W − = 0) if and only if the scalar curvature vanishes.
The result for neutral pseudo-Kähler manifolds is probably known and relates to representation theory (see [4] 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 pseudo-metric 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 Λ 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 Λ 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 Λ 2 T N : so that Λ ± is generated by E ± 1 , E ± 2 , E ± 3 . The Kähler condition implies 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. 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 Λ + ⊕Λ − (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): (Again only half the coefficients are written down.) Further simplifications come from computing W, and using First prove that the Hodge star commutes with W by considering W(Λ + , Λ − ): That proves that W is block-diagonal.
The W − term satisfies using the first Bianchi identity (and the invariance of Rm): Finally, (and indeed this matrix is traceless w.r.t. the pseudo-metric g). One should note that the above expression differs from the Riemannian case, where 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 A.2.
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: