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Almost K\"ahler manifolds whose antiholomorphic sectional curvature is pointwise constant

It is proved that if an almost K\"ahler manifold of dimension greater or equal to 8 is of pointwise constant antiholomorphic sectional curvature, then it is a complex space form.

Another characterization of the p.c.a.s.c. condition can be obtained regarding the Riemannian curvature tensor as a section of the vector bundle R(M) of the algebraic curvature tensor fields on M. According to the splitting R(M) = ⊕ 1≤i≤10 W i (M) considered in [16], the formula (1.4) can be interpreted in terms of the vanishing of suitable W i -projections p i (R) of R.
Combining with the Theorem 18 in [4], one has: Proposition 1.2. Let (M, J, g) be an almost Hermitian manidold with p.c.a.s.c. Then g is an Einstein metric iff (M, g, J) has pointwise constant holomorphic sectional curvature.
The classification of the almost Hermitian manifolds with p.c.a.s.c. is still an open problem, even if nowadays several partial results are known.
In [1] V. Apostolov, G, Ganchev and S. Ivanov classify the compact Hermitian surfaces with constant antiholomorphic curvature. Moreover, they construct an example of conformal Kähler surface with p.c.a.s.c. ν, the function ν being nonconstant. Thus, the Schur's lemma of antiholomorphic type is not valid in the 4-dimensional case.
In fact, any connected R 3 -manifold M with p.c.a.s.c. and dim M ≥ 6 has constant antiholomorphic sectional curvature ( [9]) and turns out to be a real space-form or a complex space-form ( [10]).
This result alows the classification of nearly Kähler as well as locally conformal Kähler manifolds with p.c.a.s.c. In fact, any nearly Kähler manifold is a R 3manifold ( [7]). Since for a locally conformal Kähler manifolds the projections p 9 (R) vanishes, the locally conformal Kähler manifolds with p.c.a.s.c. turn out to be R 3 -manifolds ( [3]).
Moreover, combining the results stated in [10] and [13], any connected R 3almost Kähler manifold with p.c.a.s.c. and dim M ≥ 6 turns out to be a complex space-form.
Since the projection p 9 (R), a priori, does not vanish in the almost Kähler case the classification of the almost Kähler manifolds with p.c.a.s.c is meaningful.
We recall the almost Kähler condition, i.e.: σ denoting the cyclic sum and ∇ω the covariant derivative of the fundamental 2-form ω (ω(X, Y ) = g(JX, Y )) with respect to the Levi-Civita connection ∇. Moreover, (1.8) implies: for any local orthonormal frame {e i } 1≤i≤2n . In this paper we state the following theorem, whose proof is divided into several steps. Theorem 1. Let (M, g, J) be a 2n-dimensional, n ≥ 4, connected, almost Kähler manifold. If (M, g, J) has pointwise constant antiholomorphic sectional curvature, then (M, g, J) is a complex space-form.

-Some auxiliary lemmas
Given a 2n-dimensional almost Hermitian manifold (M, g, J), the tensor field is, in general, neither symmetric nor skew-symmetric, since ρ(R), ρ * (R − L 3 R) respectively determine its symmetric, skew-symmetric components. Moreover, we assume that (M, g, J) has p.c.a.s.c.; then the formula (1.6) implies: and thus one has: for any local orthonormal frame {e i } 1≤i≤2n . If (g, J) is an almost Kähler structure, (1.8) and (1.9) imply also Now we observe that (2.1), (1.6) and (1.7) allow to rewrite (1.4) as follows: By means of (2.8) and the second Bianchi identity, we will state some properties of Q and ∇Q useful for the proof of the Theorem 1.
First of all, from (2.8), one has: where {e i } 1≤i≤2n is a local orthonormal frame.
Lemma 2.2. In the hypothesis of the Lemma 2.1, when n = 3, one has: Proof. In fact, the second Bianchi identity and (2.1) give: (2.14) Proof. The Lemmas 2.1 and 2.2, the formula (2.4) and the condition: imply the vanishing of the tensor field S defined by: In particular, by means of (1.9) and (2. turn out to be equivalent to: Thus, if n = 3, the statement follows from (2.15) combined with the condition S = 0. If n > 3, (2.16) implies also, with suitable change of the involved variables, the relation: Thus, applying (2.17) and (2.16), the relation (2.15) yields to: Moreover, via (2.17), (2.18) and (2.16), with a direct computation, one has: Therefore, the vanishing of S implies the statement.
Proposition 2.2. In the hypothesis of the Lemma 2.1, if n ≥ 4, one has: Proof. We consider the (0,3)-tensor field T such that: . Since T satisfies: T can be regarded as a section of the vector bundle W(M) whose fibre, at any point x of M, is the linear space W x considered in [8].

-The proof of the Theorem 1
To the Riemannian curvature of a manifold satisfying the hypothesis of the Theorem 1, we apply the second Bianchi identity in the form: The complete expression of the first member in (3.1), evaluated by means of (2.9), is a tensor field which contains four blocks of terms, respectively depending on g ⊗ (∇Q + Q(., ∇J)), ∇ω ⊗ Q, dν ⊗ (π 1 − 2n−1 3 π 2 ), g ⊗ ∇ω.
Since (g, J) is an almost Káhler structure, the whole term in g ⊗ ∇ω vanishes, while only the skew-symmetric component of Q, i.e. ρ * (R − L 3 R), is involved in the block depending on ∇ω ⊗ Q.
After a quite long computation, applying the Lemmas 2.1 and 2.2 and then the Proposition 2.2, the whole expression in g ⊗ (∇Q + Q(., ∇J)) turns out to depend only on dν ⊗ g ⊗ g. Thus, the condition (3.1) is equivalent to First of all, this formula implies that ν is a constant function. Indeed, given a vector field V, let Y be a vector field such that g(V, Y ) = g(JV, Y ) = 0 in an open set. Putting in (3.2) X = Z = JV , W = Y , one has: Therefore, one has: dν = 0; hence, since M is connected, ν is a constant function. Now, the condition (3.2) turns out to be equivalent to the vanishing of the tensor field H defined by: This implies also the vanishing of the tensor field H ′ defined by: Therefore, ρ * (R − L 3 R) vanishes. According to [16], this means the vanishing of the projection p 9 (R). Since also p 8 (R) = p 10 (R) = 0 (see also the Proposition 1.1), (M, g, J) turns out to be a R 3 -almost Kähler manifold with p.c.a.s.c. Since dim M ≥ 8, a direct application of the classification theorem in [10] implies that (M, g, J) is a Kähler manifold with constant holomorphic sectional curvature. This contradicts the condition ∇J = 0.