Rigidity results for Riemannian twistor spaces under vanishing curvature conditions

In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that CP3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}\mathbb {P}^3$$\end{document} is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold.


Introduction and main results
Let (M, g) be an oriented Riemannian manifold of dimension 2n, with metric g.The twistor space Z associated to M is defined as the set of all the couples (p, J p ) such that p ∈ M and J p is a complex structure on T p M compatible with g, i.e. such that g p (J p (X), J p (Y )) = g p (X, Y ) for every X, Y ∈ T p M . 4lternatively, we can define Z in an equivalent way as where O(M ) denotes the orthonormal frame bundle over M and the unitary group U (n) is identified with a subgroup of SO(2n) (see [13] for further details).These structures, introduced by Penrose ( [25]) as an attempt to define an innovative framework for Physics, have been the subject of many investigations by the mathematical community, in virtue of the numerous geometrical and algebraic tools involved in the definition of their properties.In 1978, Atiyah, Hitchin and Singer ( [1]) adapted Penrose's twistor theory to the Riemannian context, introducing the concept of twistor space associated to a Riemannian four-manifold and paving the way for many researches about this subject.
The orientation on M implies that O(M ) has two connected components, O(M ) + and O(M ) − , and therefore we can define the two connected components of Z where SO(M ) is the orthonormal oriented frame bundle over M .We choose the component Z − to be the twistor space of (M, g) (see also [5] and [27]).It is possible to define a natural family of Riemannian metrics g t on Z − , where t > 0 ( [13] [22]); from now on, we systematically use the notation (Z, g t ) to denote the twistor space Z − endowed with the Riemannian metric g t .
In general, if (M, g) is a Riemannian manifold of dimension m ≥ 3, the Riemann curvature tensor Riem on M admits the well known decomposition where W, Ric and S denote the Weyl tensor, the Ricci tensor and the scalar curvature of M , respectively, and ∧ is the Kulkarni-Nomizu product.Moreover, the Riemann curvature tensor defines a symmetric linear operator from the bundle of two-forms Λ 2 to itself where {θ i } i=1,...,m is a local orthonormal coframe on an open set U ⊂ M , with dual frame {e i } i=1,...,m , γ kt = γ(e k , e t ) and R ijkt are the components of the Riemann tensor with respect to the coframe {θ i }.
If m = 4 and M is oriented, Λ 2 splits, via the Hodge ⋆ operator, into the direct sum of two subbundles Λ + and Λ − .This implies that the Riemann curvature operator R assumes a block matrix form The paper is organized as follows: in Section 2, we show that, given a Riemannian four-manifold (M, g), its twistor space (Z, g t ) cannot be locally conformally flat for any t > 0. Section 3 is devoted to the characterization of Bochner-flat twistor spaces: in particular, we show that the only Bochner-parallel twistor space is "essentially" CP 3 , which is the one associated to the four-sphere S 4 .
In Section 4 we consider Ricci parallel and locally symmetric twistor spaces, providing rigidity results for twistor spaces whose Atiyah-Hitchin-Singer almost complex structure J + = J is integrable (see Appendix C for details).
In Section 5 we prove a general quadratic formula for |∇J| 2 ; moreover, we generalize the necessary and sufficient condition for the integrability of J, first proven by Atiyah, Hitchin and Singer [1], through a vanishing condition on the divergences of the associated Nijenhuis tensor.We also prove a new result concerning the Nijenhuis tensor of the Eells-Salamon almost complex structure J − = J (see [15]).
To keep the paper self-contained as much as possible, we provide also five brief appendices devoted to technicalities and some heavy computations (for instance, the list of the local components of the Weyl tensor of a twistor space (Z, g t )).

Locally conformally flat twistor spaces
In this section, we want to show that the twistor space (Z, g t ) associated to a Riemannian four-manifold (M, g) is never locally conformally flat for any t > 0. By Weyl-Schouten Theorem, we know that a Riemannian manifold of dimension n ≥ 4 is locally conformally flat if and only if its Weyl tensor W vanishes identically (for a proof, see [20] or [23]).
Before we state the main result of this section, let us recall the transformation laws for the matrices A and B appearing in the decomposition of the Riemann curvature operator: we know that, given a local orthonormal frame e ∈ O(M ) − , if we choose another frame e ∈ O(M ) − , the change of frames is determined by a matrix a ∈ SO (4) and that the matrices A and B transform according to the equations (2.1) where SO(3)× SO(3) ∈(a + , a − ) = µ(a) and µ is a surjective homomorphism from SO(4) to SO(3)× SO(3) induced by the universal covers of SO(4) and SO(3) (see [2], [5] and [27] for a detailed description).
For the sake of simplicity, throughout the paper we adopt the following notation We also compute the differentials of the components listed in (2.2): where {ω 1 , ..., ω 4 } is a local orthonormal coframe and ω i j are the associated Levi-Civita connection 1-forms.Now, we can state the following result, which is new, to the best of our knowledge: Theorem 2.1.Let (M, g) be a Riemannian four-manifold and (Z, g t ) be its twistor space.Then, (Z, g t ) is not locally conformally flat for any t > 0.
Proof.Let us suppose that (Z, g t ) is locally conformally flat, i.e., by Weyl-Schouten Theorem, W ≡ 0 on Z.By the vanishing of the coefficients W ab56 in (B.4), we obtain the system expliciting the right-hand sides and then summing the equations, we derive the equality Note that W ≡ 0 is a global condition: in particular, this means that the equation above must hold for every p ∈ M (it suffices to consider the pullback maps via any section of the twistor bundle).Moreover, since the locally conformally flatness is a frame-independent condition, the equation holds for every local negatively oriented orthonormal frame e ∈ O(M ) − .In particular, since A is a symmetric matrix, we have that the equality holds for every frame e such that A is diagonal; in this situation, we have that for every frame with respect to which A is diagonal.By (2.1), we can exchange the diagonal entries of A with suitable changes of frames in order to obtain the additional equations 0 where A ij and A ij are the entries of the matrix A with respect to some frames e and e, respectively.At a point p ∈ M , since t > 0, the system of these three equations admits three distinct solutions: (1) A 11 = A 22 = A 33 = 0; (2) (3) two diagonal entries out of three are equal to − 2 t 2 , while the third is equal to 2 t 2 .This means that, at p ∈ M , the scalar curvature S of (M, g) can attain the values 0, 24/t 2 or −8/t 2 .Since the scalar curvature is a smooth function on M and, for every point of M , one of the three equations must hold, we can conclude that S is constant on M : indeed, the possible values for S are finitely many, therefore, if S(p) = S(p ′ ) for p, p ′ ∈ M , S would not be a smooth function.
First, let us prove that the first two cases lead to a contradiction.Note that, in this situation, A is a scalar matrix for every point p ∈ M (and, by (2.1), for every frame), which means that (M, g) is a self-dual manifold.By the vanishing of the components W 5ab5 and W 6ab6 , if a = b, we obtain 4), by the self-duality condition we can compute = 0 on M .This is equivalent to say that the entries B 23 and B 32 of the matrix B vanish identically on M ; since this is a global condition, by suitable change of frames, equation (2.1) implies that the matrix B is the zero matrix, i.e. (M, g) is an Einstein manifold (see also [5] for a detailed proof).However, we have that the left-hand side of the second equation is equal to (S 2 /18) for an Einstein, self-dual manifold, hence, for S equal to 0 or to 24/t 2 , we get a contradiction.Thus, we can choose a frame e with respect to which A is diagonal and If we consider again the equations W 5ab5 = W 6ab6 = 0, we obtain = 0, i.e.B 32 = B 23 = 0 in the chosen frame e.In fact, we can say more: the equalities B 32 = B 23 = 0 hold for every frame e ′ with respect to which the matrix A is in diagonal form with A 11 = −A 22 = A 33 = −2/t 2 .Note that we can choose suitable change of frames such that A = A, where A is the matrix associated to the transformed frame e: indeed, it suffices to choose a + = I 3 in (2.1).
Therefore, with suitable choices of a − and putting a + = I 3 in (2.1), it is immediate to show that for a frame e with respect to which A is in diagonal form with Finally, let us compute , which is obviously impossible.Thus, (Z, g t ) cannot be locally conformally flat.

Bochner-flat twistor spaces
Let (N, g, J) be a almost Hermitian manifold of dimension 2n.We can define the Bochner tensor B of N as the (0, 4)-tensor whose components with respect to a local orthonormal frame are δ ps δ qr − δ pr δ qs + J p s J q r − J p r J q s − 2J p q J r s .
This tensor was first introduced by Bochner as a "complex analogue" of the Weyl tensor [3].It is important to note that some authors define the Bochner tensor as − B, because of a different convention for the sign of the Riemann tensor (see, for instance, [31] and [32]).
We say that N is a Bochner-flat manifold if B vanishes identically, i.e. if B pqrs = 0 for every 1 ≤ p, q, r, s ≤ 2n.It is known that, in general, the Bochner tensor does not satisfy the same symmetries as the Riemann tensor (see, for instance, [35]).However, if N is Bochner-flat, by (3.1) we obtain δ ps δ qr − δ pr δ qs + J p s J q r − J p r J q s − 2J p q J r s , which means that the right-hand side of (3.2) satisfies the same symmetries as the Riemann tensor.Now, let (M, g) be a four-dimensional Riemannian manifold and let (Z, g t , J) be its twistor space, regarded as an almost Hermitian manifold.It is known that (Z, g t , J) is a Kähler-Einstein manifold if and only if (M, g) is an Einstein, self-dual manifold with scalar curvature S = 12/t 2 (see, for instance, [5], [10], [24] and Corollary 4.2 of this paper).Let us suppose that (Z, g t , J) is a Kähler-Einstein manifold and let B be its Bochner tensor.Under these hypotheses, we can compute the components B pqrs : It is natural to ask whether Proposition 3.1 can be generalized or not if there are no hypothesis on the almost complex structure J: more precisely, our goal is to characterize almost Hermitian, Bochner-flat twistor spaces.Rather surprisingly, it turns out that S 4 is the only Riemannian four-manifold whose twistor space is Bochner-flat.
First, let us define the covariant derivative ∇B of the Bochner tensor B of an almost Hermitian manifold (N, g, J), whose components with respect to a local orthonormal coframe are and We say that (N, g, J) is Bochner-parallel if ∇ B ≡ 0; in this case, by (3.4) the components of ∇ Riem satisfy the equation Before we state and prove the main result of this section, we need the following Theorem 3.2.Let (M, g) be a Riemannian four-manifold and (Z, g t , J) be its twistor space.Then, the Ricci tensor Ric of Z is complex linear, i.e.
if and only if (M, g) is an Einstein, self-dual manifold.
Proof.If (M, g) is an Einstein, self-dual manifold, the validity of (3.6) can be immediately shown by a direct inspection of the components listed in (B.2).Thus, let us suppose that (3.6) holds on Z. First, note that on Z; by (B.2), we can write ).
Subtracting the first equation from the second, we obtain note that this is a global condition on the entries of the matrix B, which means that it holds for every choice of local orthonormal frame.By (2.1), we can choose where I 3 is the 3 × 3 identity matrix, to compute and to obtain

Now, choosing
therefore, we conclude that B 22 = B 32 = 0. Now, we can choose the change of frames determined by which obviously implies that B 23 = B 33 = 0.By an analogous computations, we can also obtain B 21 = B 31 = 0. Finally, choosing and repeating the argument above, we conclude that B 11 = B 12 = B 13 = 0. Thus, for every p ∈ M there exists a local frame such that B = 0, i.e. (M, g) is an Einstein manifold.Now, by (3.6), we obtain since (M, g) is an Einstein manifold, this equation can be rewritten as which is another global condition and, therefore, does not depend on the choice of the local frame.In particular, for every frame with respect to which A is diagonal, we have that (A 22 ) 2 = (A 33 ) 2 and, since we can exchange the diagonal entries of A, we can conclude If A 11 = A 22 = A 33 , then this holds for every local orthonormal frame by (2.1), hence (M, g) is self-dual and the claim is proven.Thus, without loss of generality, we may suppose that, for instance, A 22 = A 33 at a point p ∈ M for some local frame e ∈ O(M ) − with respect to which A is diagonal.By the equation above, it is obvious that the diagonal entries satisfy if we choose the change of frames determined by we obtain which is true if and only if x = 0 and leads to a contradiction.Therefore, diagonalizing A we obtain a scalar matrix on M , which is equivalent to say that (M, g) is a self-dual manifold.
We are now ready to state the following Theorem 3.3.Let (M, g) be a Riemannian four-manifold and (Z, g t , J) be its twistor space.Then, (Z, g t , J) is Bochner-parallel if and only if (M, g) is homotetically isometric to S 4 with its canonical metric.In particular, the only Bochner-parallel twistor space is CP 3 endowed with the Fubini-Study metric.
An immediate consequence of Theorem 3. Let us consider the local expression of ∇ B in (3.5): the right-hand side satisfies the same symmetries as the components of ∇ Riem.Thus, recalling that the Riemann tensor is skew-symmetric with respect to the last two indices, we obtain for every p, q = 1, ..., 6.
If we consider a pair of indices (r, s) such that J r s = 0, the equation becomes J r s,u (R pt J t q + R qt J t p ) = 0.If J r s,u = 0 for some r, s, u and for every local frame e, we conclude that R pt J t q + R qt J t p = 0 for every p, q.On the other hand, if e ′ is a frame with respect to which J r s,u = 0 for every r, s, u, we obtain If we suppose S = 0, by (B.1) and (3.5) it is easy to compute which is impossible, since Thus, by Corollary 4.2 we conclude that (Z, g t , J) is a Kähler-Einstein manifold and this ends the proof.
(1) It is worth to note that Theorem 3.2 is a generalization of a result due to Davidov, Grantcharov and Muškarov, who showed that the Riemann tensor Riem of (Z, g t , J) satisfies , s ≤ 6 if and only if (M, g) is an Einstein, self-dual manifold (see [12]).Indeed, a straightforward computation shows that (3.7) implies (3.6).Almost Hermitian manifolds which satisfy (3.7) are sometimes called RKmanifolds see ( [33] and [35]) and (3.7) is a condition satisfied by every nearly Kähler manifold (see [18]).(2) We point out that one can directly prove Corollary 3.4 without exploiting Theorem 3.3; indeed, if we suppose that (Z, g t , J) is a Bochner-flat manifold, by (3.2) we can show that Riem is a K-curvature-like tensor, i.e. its components satisfy (3.8) R pqts J t r + R pqrt J t s = 0, ∀1 ≤ p, q, r, s ≤ 6.The equality in (3.8) is sometimes referred to as Kähler identity and it is a deeply studied feature of almost Hermitian manifolds, aside from the twistor spaces context (we may refer the reader to [18], [28], [29], [34] and [35]).

Ricci parallel and locally symmetric twistor spaces
In this section, we discuss the case of a Riemannian four-manifold (M, g) whose twistor space (Z, g t , J) is a Kähler-Einstein manifold.
Let us start proving a well-known result due to Friedrich and Grunewald (see [16]): We obtain immediately that R a5 = R a6 = 0: indeed, since M is Einstein, then (M, g) has a harmonic curvature metric, i.e. div Riem = 0, which means that, in particular, Q ac . Therefore, for instance we have that with similar computations, we obtain that R ab = 0 for every a = b.Obviously, the system also implies that R 56 = 0. Now, let us consider 3 we have that 72 Thus, we have This implies immediately that Subtracting the two equations, it is easy to show that . Now, by the expression of the components in (B.2), we have that a straightforward computation shows that that is, for every local orthonormal frame e ∈ O(M ) − .Thus, we can choose a frame e such that the associated matrix A is diagonal, in order to obtain B 11 = 0 (note that this is not a global condition).By (2.1), if we us choose we easily obtain that (note that also A is a diagonal matrix).By analogous computations, we can conclude that B = 0, i.e. (M, g) is an Einstein manifold.Now, by hypothesis we have that since (M, g) is Einstein, this equation assumes the form Let us again choose e ∈ O(M ) − such that A is diagonal.Choosing suitable matrices a + ∈ SO(3), it is easy to obtain that If we suppose that A 11 = A 22 , analogous computations immediately lead to a contradiction.Thus, A 11 = A 22 .
Exchanging A 22 and A 33 , we can repeat the same argument to obtain A 11 = A 33 .Thus, (M, g) is self-dual.Finally, since (M, g) is Einstein and self-dual, we compute Again, this equation holds if and only if 12 t 2 and this ends the proof.
As a consequence, we can state another well known result (see [24] and [5]): Corollary 4.2.The twistor space (Z, g t , J) associated to (M, g) is a Kähler-Einstein manifold if and only if (M, g) is an Einstein, self-dual manifold with scalar curvature equal to 12/t 2 .
Proof.If (M, g) is an Einstein, self-dual manifold with S = 12/t 2 , a direct inspection of the components in (B.2) and (C.3) shows that (Z, g t ) is Kähler-Einstein.Conversely, by Theorem 4.1 we know that (M, g) is Einstein, self-dual with S equal to 6/t 2 or 12/t 2 .If we suppose S = 6/t 2 , we immediately obtain that which contradicts the hypothesis that (Z, g t , J) is a Kähler manifold and ends the proof.
We recall that compact, Einstein, self-dual four-manifolds with positive scalar curvature have been classified: indeed, Hitchin showed that there are just two possibilites, up to conformal equivalences, which are S 4 and CP 2 with their canonical metrics [21].Now, we want to provide an analogue of Proposition 3.1, in order to give another characterization of S 4 .Indeed, by direct inspection of the components listed in (B.6), one can immediately show the validity of Theorem 4.3.Let (Z, g t , J) be a Kähler-Einstein twistor space.Then, (Z, g t , J) is locally symmetric if and only if (M, g) is homotetically isometric to S 4 with its canonical metric.
Note that, combining Theorem 4.3 with Theorem 3.3 and Corollary 2.2, we can state the following Theorem 4.4.Let (Z, g t , J) be a Kähler-Einstein twistor space.Then the following conditions are equivalent: (1) (Z, g t , J) is a conformally symmetric manifold; (2) (Z, g t , J) is a Bochner-parallel manifold; (3) (M, g) is homotetically isometric to S 4 , with its canonical metric.
Recall that the equivalence (2) ⇔ (3) holds without any a priori hypothesis on (Z, g t , J), by Theorem 3.3.
Moreover, the Einstein condition on g t implies that (Z, g t , J) is a harmonic curvature manifold, i.e. div Riem ≡ 0.Then, by this consideration, equation (3.4) and Theorem 4.4 we can state the following Theorem 4.5.Let (Z, g t , J) be a Kähler-Einstein twistor space.Then, either (Z, g t , J) is Bochner-parallel (and, then, (M, g) is homotetically isometric to S 4 ) or ∇B = 0 and div B ≡ 0.
Though in Theorem 3.3 we characterized S 4 as the only four-dimensional manifold whose twistor space is Bochner-parallel, we cannot obtain an analogous characterization if we drop the hypothesis of g t being Kähler-Einstein in Theorem 4.3.For instance, by (B.1), an easy computation of the local expression of ∇Riem proves the following Proposition 4.6.Let (M, g) be a Ricci-flat, self-dual, locally symmetric four-manifold.Then, its twistor space (Z, g t ) is locally symmetric.
Even though S 4 is not the only four-manifold whose twistor space satisfies ∇Riem ≡ 0, if we consider Hermitian twistor spaces, i.e. the ones associated to self-dual manifolds (see [1] and [5]), we can state the following Theorem 4.7.Let (M, g) be a self-dual Riemannian manifold and let (Z, g t ) be its twistor space.Then, (1) (Z, g t ) is Ricci parallel (i.e., ∇Ric ≡ 0) if and only if (Z, g t ) is an Einstein manifold or (M, g) is Ricci-flat; (2) (Z, g t ) is locally symmetric if and only if (M, g) is homotetically isometric to S 4 with its canonical metric or (M, g) is Ricci-flat and locally symmetric. Proof.
(1) First, let us suppose that (M, g) is an Einstein, self-dual manifold with scalar curvature S. In particular, (M, g) is Ricci parallel.Moreover, since under these hypotheses for every a, b, c, we immediately obtain that the only components listed in (B.7) that may not vanish are R ab,5 , R ab,6 , R a5,b , R a6,b for some a, b.A straightforward computation shows that R ab,5 = R ab,6 = 0, regardless of the value of S.
Let us consider, for instance, Similar computations show that the other components of ∇Ric vanish for the same values of S. Thus, by Theorem 4.1, we conclude that, if (M, g) is an Einstein, self-dual manifold, (Z, g t ) is Ricci parallel if and only if it is an Einstein manifold or the scalar curvature of (M, g) vanishes.
Thus, in order to prove the statement, it is sufficient to show that any self-dual manifold whose twistor space is Ricci parallel must also be an Einstein manifold.Under these hypotheses, we can observe that The validity of these three equations is equivalent to the orthogonality of the columns of the matrix B; this means that B T B = D, where D is a diagonal matrix.Moreover, let us denote the i-th column of B as v i and let us define furthermore, let us suppose that v i = 0 for every i.Since the columns of B are the rows of B T , if we replace the rows of B T with v T i /||v i || 2 , we obtain that the new matrix, which we denote as B or , is orthogonal.In particular, we may assume that B or ∈ SO(3) (otherwise, we could replace it with −B or ); thus, putting a −1 − = B or and a + = I 3 in (2.1), we have that By the expressions of the entries of the matrices A and B, it is easy to obtain The two equations can hold simultaneously if and only if t = 3 + √ 17, which implies that However, since S is invariant under change of frames, we can choose now, it is easy to compute R 12,5 = −t + t 3 S 12 = 0, which holds if and only if Therefore, we get a contradiction and we conclude that at least one of the columns of B must be made of zeros for every e ∈ O(M ) − , which obviously implies that (M, g) is an Einstein manifold.
(2) We know that S 4 and every locally symmetric, self-dual, Ricci-flat manifold have locally symmetric twistor spaces, by Proposition 4.6.Conversely, let us suppose that (Z, g t ) is locally symmetric: in particular, it is Ricci parallel, therefore (Z, g t ) is an Einstein manifold or (M, g) is Ricci-flat.
If S = 0, by (B.1), it is easy to compute it is apparent that (M, g) is locally symmetric by hypothesis and this ends the proof.

5.
A quadratic formula for ∇J and higher order conditions 5.1.General quadratic formula for the square norm of ∇J.We begin stating a general result: Theorem 5.1.Let (M, g) be a Riemannian four-manifold and (Z, g t , J) be its twistor space.Then, the equality holds, where |dω| 2 = 6 p,q,t=1 dω(e p , e q , e t )dω(e p , e q , e t ) and |N J | 2 = 6 p,q,t=1 N t pq N t pq .
Proof.By direct computation, we obtain Comparing these expressions with (C.4), it is easy to obtain (5.1).
It is worth to point out that Theorem 5.1 allows to give alternate proofs of some well-known results due to Muškarov (see [24]), exploiting the quadratic relations among the invariants listed by Gray and Hervella (see [19]).For instance, we can reformulate the following Proposition 5.2.Let (M, g) be a Riemannian manifold and (Z, g t , J) be its twistor space.Then, (Z, g t , J) ∈ N K ∪ AK ⇔ (Z, g t , J) ∈ K, where N K, AK and K denote the classes of nearly Kähler, almost Kähler and Kähler manifolds, respectively.
Proof.One direction is trivial.Let us suppose Z ∈ N K.Then, by table IV in [19], we know that inserting this equation in (5.1), it is easy to obtain that is, Z ∈ K. Now, let us suppose Z ∈ AK.By the same table, we have that again, inserting these equations in (5.1), we obtain |∇J| 2 = 0, i.e.Z ∈ K.
In fact, by analogous calculations, we can prove more.Let us consider the sixteen classes of almost Hermitian manifolds listed in [19].Then, by Theorem 5.1, we can obtain an alternate proof of the following statement, which was first proven by Muškarov (see [24]): Theorem 5.3.Let (M, g) be a Riemannian four-manifold and (Z, g t , J) be its twistor space.If (Z, g t , J) belongs to one of the first fifteen classes of almost Hermitian manifolds, then (Z, g t , J) ∈ H; consequently, (M, g) is self-dual.

Laplacian of the almost complex structures.
In this section, we consider the Laplacian ∆ J J of the almost complex structure J (for the definition and the components, see D).We say that J is harmonic if ∆ J J = 0 (see also [37] and [38]).By a direct inspection of the components listed in D, we can provide an alternate proof to a well-known result, due to Davidov and Muškarov (see [11]): Theorem 5.4.Let (M, g) be a Riemannian four-manifold and (Z, g t , J) be its twistor space.Then, J is harmonic if and only if (M, g) is self-dual.
Proof.One direction is trivial; indeed, since (M, g) is self-dual, where the N r pq are the components of the Nijenhuis tensor of J (see (E.1)).Moreover, as an immediate consequence of the self-duality condition and the second Bianchi identity, by (A.1) and (D.2) it is easy to show that Conversely, let us suppose ∆ J J = 0.By the explicit expression of ∆ J J u v and N r pq listed in (D.1) and in (E.1), respectively, we obtain the global equations Again, let us choose a local orthonormal frame e ∈ O(M ) − such that A is a diagonal matrix.By the transformation law for the matrix A defined in (2.1), choosing the matrix and using the equations (5.2), we obtain Let us suppose that A 11 = A 22 ; this implies immediately that With similar computations, it is easy to show that, if this equation holds, by (5.2) we must have In both cases, choosing the matrix and applying the transformation law for A, the equations (5.2) lead to which clearly are contradictions.Thus, A 11 = A 22 ; by analogous computations, we easily obtain A 22 = A 33 .Therefore, we can conclude that (M, g) is self-dual.

5.3.
Nijenhuis tensors of J and J. Now, let us consider the Nijenhuis tensors N J and N J associated to J and J respectively.The expression of the components J p q,t and J p q,t of the covariant derivatives of J and J are listed in (C.3) and (C.5), while the components of the tensors N J and N J are listed in (E.1) and (E.6), together with the components of their covariant derivatives and their divergences in (E.3) and (E.7).Thus, let us recall the local expression of the divergences of N J with respect to a local orthonormal coframe and its dual frame: where ∇N J = N p tq,s θ s ⊗ θ t ⊗ θ q ⊗ e p .The following Theorem is a generalization of a result due to Atiyah, Hitchin and Singer, which characterizes self-dual four-manifolds as the ones whose twistor space is Hermitian with respect to J (see [1] and [5]): Theorem 5.5.Let (M, g) be a Riemannian four-manifold and (Z, g t , J) be its twistor space.Then, (M, g) is self-dual if and only if div N J ≡ 0 ∨ divN J ≡ 0.
Proof.One direction is trivial: indeed, we know that, if (M, g) is self-dual, N J ≡ 0. Therefore, it is easy to see that div N J = divN J ≡ 0.
Conversely, let us suppose that div N J ≡ 0. By (E.1) and (E.4), we have Let us choose a frame e ∈ O(M ) − such that A is diagonal.By (2.1), choosing for the transformed matrix we obtain that is, and applying again (2.1), it is easy to see that A 11 = A 22 , which is a contradiction.Thus, with respect to e we have that A 11 = A 22 .By a similar argument, one can easily show that A 11 = A 33 , i.e. (M, g) is self-dual.Now, let us consider the case divN J ≡ 0 and let us choose again a frame e ∈ O(M ) − such that A is diagonal.Then, since by hypothesis N 1 1t,t + N 2 2t,t = 0, we can rewrite this equation as (A 33 − A 22 ) 2 = 0, which means that A 22 = A 33 .The equation above holds for every frame with respect to which the matrix A is diagonal; thus, similarly we can obtain As a consequence, it is immediate to show the validity of Corollary 5.6.(M, g) is self-dual if and only if ∇N J ≡ 0.
We point out that the equation ∇N J ≡ 0 has been studied in the wider context of almost Hermitian manifolds: for instance, Vezzoni showed that any almost Kähler manifold that satisfies this condition is, in fact, a Kähler manifold (see [36]).
Note that, on the contrary, a simple inspection of the coefficients listed in (E.7) shows that ∇N J never vanishes (the fact that J is never integrable was first proven by Eells and Salamon [15] and it is apparent by (E.6); we also mention that, if (M, g) is Einstein and self-dual, N J is parallel with respect to the Chern connection ∇ defined on (Z, g t , J), as shown by Davidov, Grantcharov and Muškarov [9]); thus, we cannot obtain an analogue of Corollary 5.6 for N J .However, we can consider the divergences div N J and divN J of N J (the components are listed in (E.8) and (E.9)) and state the following Theorem 5.7.Let (M, g) be a four-dimensional, self-dual Riemannian manifold.If M is Ricci-flat, then divN J = 0.If the scalar curvature S of (M, g) is different from 6/t 2 , then the converse holds.
Proof.Suppose that (M, g) is Ricci-flat, i.e.Ric = 0.In particular, (M, g) is Einstein, self-dual with S = 0.This implies that In particular, by the second equation we obtain Therefore, divN J = 0, by direct inspection.Now, let us suppose that divN J = 0 and S = 6/t 2 .It is easily shown that the hypothesis on the scalar curvature leads to Σ − 2 t 2 = 0, on O(M ) − .
Indeed, if Σ − 2 t 2 = 0, the matrix A appearing in the decomposition of the Riemann curvature operator has the form for every local orthonormal frame, since M is self-dual.Then, we must have , which is a contradiction.Thus, by hypothesis, we must have These equations immediately imply that (M, g) is Einstein with S = 0, i.e.M is Ricci-flat.

Appendix A. Divergence of the self-dual part of the Weyl tensor
We list the components of the divergence of W + in a Riemannian four-manifold.
Let (Z, g t , J) be a Kähler-Einstein manifold.By (B.1), the components of the Riemann tensor Riem of Z are (3,4); (3,4); The components of ∇Riem are R abcd,e = R abcd,e , R abcd,5 = R abcd,6 = R abc5,5 = R abc5,6 = R abc6,5 = R abc6,6 = 0; The general expressions of the components of ∇Ric are The covariant derivative of the scalar curvature S has components Appendix C. Covariant derivative of the almost complex structures and differential of the Kähler forms In this appendix we list all the components of the covariant derivative of the almost complex structures J ± on Z. Recall that using the same notation of the article, we write Kähler forms of J and J: denoting by ω + and ω − the Kähler forms of J and J, respectively, we have: As far as the codifferentials of ω + and ω − are concerned, we have: q,t θ t r + J t q,r θ p t ; J p q,rt = −J q p,rt .J 4 5,rt = J 3 6,rt ; J 4 6,rt = −J 3 5,rt ; J 2 3,rt = J 1 4,rt ; J 2 4,rt = −J 1 3,rt ; J 2 5,rt = J 1 6,rt ; J 2 6,rt = −J 1 5,rt .
The local components of the Laplacian ∆ J J are (D.1) ∆ J J u v := (∆J − J∇ p J∇ p J) u v = ∆J u v − J u t J t q,p J q v,p .

3 ,c = Q ac 4 ,
c = 0 for every a = 1, ..., 4. We have that M is Einstein and self-dual if and only if
Again, the right-hand side is equal to S/6 if and only if S is equal to 6/t 2 or to 12/t 2 , which implies that Z is Einstein.Conversely, let us suppose that (Z, g t ) is an Einstein manifold, i.e.R pq = (S/6)δ pq .Recalling the expressions of Riem, Ric and S listed in (B.1), (B.2) and (B.3), respectively, we easily obtain i.e.R 11 = S/6 by hypothesis.With analogous computations, we conclude that R aa = S/6 for every a.Finally,R 55 = R 66 = 1 t 2 + t 2 S 2 144 .
Since the system holds for every e ∈ O(M ) − , by a suitable change of frames we obtain thatB 12 B 13 + B 22 B 23 + B 32 B 33 = 0.