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The fundamental form of almost-quaternionic Hermitian manifolds

We prove that if the fundamental 4-form of an almost-quaternionic Hermitian manifold (M, Q, g) of dimension at least eight satisfies the conformal-Killing equation, then (M, Q, g) is quaternionic-Kahler.


Introduction
Conformal-Killing (respectively, Killing) 1-forms are dual to conformal-Killing (respectively, Killing) vector fields. More generally, a p-form ψ (p ≥ 1) on a Riemannian manifold (M m , g) is conformal-Killing, if it satisfies the conformal-Killing equation where ∇ is the Levi-Civita connection and (like everywhere in this note) we identify tangent vectors with 1-forms by means of the Riemannian duality. Co-closed conformal-Killing forms are called Killing. Note that ψ is Killing if and only if its covariant derivative is totally skew, or, equivalently, (∇ X ψ)(X, ·) = 0 for any vector field X.
Conformal-Killing forms exist on spaces of constant curvature, on Sasaki manifolds [6] and on some classes of Kähler manifolds, like Bochner-flat Kähler manifolds and conformally-Einstein Kähler manifolds [1], [4]. On compact quaternionic-Kähler manifolds of dimension at least eight, there are no non-parallel conformal-Killing 2-forms, unless the quaternionic-Kähler manifold is isomorphic to the standard quaternionic projective space, in which case the space of conformal-Killing 2-forms is naturally isomorphic to the space of Killing vector fields [3].
Conformal-Killing forms exist also on manifolds which admit twistor spinors [6]. Recall that a twistor spinor on a Riemannian spin manifold (M m , g) is a section ρ of the spinor bundle, which satisfies the equation where X is any vector field, D is the Dirac operator and "·" denotes the Clifford multiplication. If ρ 1 and ρ 2 are twistor spinors, then the p-form ω p (X 1 , · · · , X p ) = (X 1 ∧ · · · ∧ X p ) · ρ 1 , ρ 2 is conformal-Killing (for any p ≥ 1). For a survey on conformal-Killing forms, see for example [6].
The starting point of this note is a result proved in [6], which states that if the Kähler form of an almost-Hermitian manifold is conformal-Killing, then the almost-Hermitian manifold is nearly Kähler. Our main Theorem is an analogue of this result in quaternionic geometry and is stated as follows: Theorem 1. Let (M 4n , Q, g) be an almost-quaternionic Hermitian manifold, of dimension 4n ≥ 8. Suppose that the fundamental 4-form Ω of (M, Q, g) is conformal-Killing. Then (M, Q, g) is quaternionic-Kähler.
Theorem 1 generalizes a result proved in [8], namely that in dimension at least eight, a nearly quaternionic-Kähler manifold (i.e. an almostquaternionic Hermitian manifold for which the fundamental 4-form is a Killing form) is necessarily quaternionic-Kähler.
The paper is organized as follows: in Section 2 we recall basic facts on quaternionic Hermitian geometry. Section 3 is devoted to the proof of our main result, which is based on a representation theoretic argument. Similar arguments were already employed in [7] and [8].

Quaternionic Hermitian geometry
Let M be a manifold of dimension 4n ≥ 8 (in all our considerations the dimension of the manifold will be at least eight). An almost-quaternionic structure on M is a rank-three vector sub-bundle Q ⊂ End(T M), locally generated by three anti-commuting almost complex structures {J 1 , J 2 , J 3 } which satisfy J 1 • J 2 = J 3 . Such a triple of almost complex structures is usually called a (local) admissible basis of Q. An almost-quaternionic Hermitian structure on M consists of an almost-quaternionic structure Q and a Riemannian metric g compatible with Q, which means that In the language of G-structures, an almost-quaternionic Hermitian structure on a 4n-dimensional manifold is an Sp(n)Sp(1)-structure. Therefore, on an almost-quaternionic Hermitian manifold (M 4n , g, Q) there are two locally defined complex vector bundles E and H, of rank 2n and 2 respectively, associated to the standard representations of Sp(n) and Sp(1) on E = C 2n and H = C 2 . Let ω E ∈ Λ 2 (E * ) and j E : E → E be the standard symplectic form and quaternionic structure of the bundle E, defined by the Sp(n)invariant complex symplectic form and quaternionic structure of E. We shall often identify E with E * by means of the map e → ω E (e, ·), so that ω E will sometimes be considered as a bivector on E. For any r ≥ 2 we shall denote by Λ r 0 E ⊂ Λ r E the kernel of the natural contraction with the symplectic form ω E , defined by ω E • (e 1 ∧ · · · ∧ e r ) = i<j (−1) i+j+1 ω E (e i , e j )e 1 ∧ · · · ∧ e i ∧ · · · ∧ e j ∧ · · · ∧ e r where the hat denotes that the term is omitted. By means of contraction and wedge product with ω E we can decompose Λ r E as The map j E is complex anti-linear and for any u, v ∈ E and e ∈ E \ {0}. To simplify notations, for a vector e ∈ E we shall often denoteẽ := j E (e) its image through the quaternionic structure of E. Similar conventions will be used for the standard symplectic form ω H ∈ Λ 2 (H * ) and quaternionic structure j H : H → H of the bundle H.
The bundles E and H play the role of spin bundles from conformal geometry. In particular, and the complex bilinear extension of the Riemannian metric g to T C M is the tensor product ω E ⊗ ω H . Decomposition (4) induces decompositions of the form bundles in any degree. In particular, the bundles of 2 and 3-forms decompose as (see [5]) (In (5) and (6), and often in this note, we omit the tensor product signs). In (5) S 2 H and S 2 E are complexifications of the bundle Q and, respectively, of the bundle of Q- In (6) K denotes the vector bundle associated to the Sp(n)-module K, which arises into the irreducible decomposition under the action of Sp(n). A vector from E ⊗ Λ 2 0 E has non-trivial component on K if and only if it is not totally skew.
Notations 2. We shall identify bundles with their complexification, without additional explanations. For example, in (5) S 2 HΛ 2 0 E is a complex subbundle of Λ 2 (T C M). We shall use the same notation for its real part, which is a sub-bundle of Λ 2 (T M).
An almost-quaternionic Hermitian manifold (M, g, Q) has a canonical 4form, defined, in terms of an arbitrary admissible basis where ω i := g(J i ·, ·) are the Kähler forms corresponding to (g, J i ). As proved in [2] and [7], the covariant derivative ∇Ω with respect to the Levi-Civita using the Riemannian metric), in the following way. Note first that Λ 2 (S 2 H) is canonically isomorphic to S 2 H (this is because S 2 H is the complexification of Q, which has a natural metric and orientation, for which any admissible basis {J 1 , J 2 , J 3 } is orthonormal and positively oriented). The map is the promised embedding of S 2 HΛ 2 0 E into Λ 4 (T M). An almost-quaternionic Hermitian manifold (M, Q, g) is quaternionic-Kähler if the Levi-Civita connection ∇ of g preserves the bundle Q, or, equivalently, the fundamental 4-form Ω is parallel with respect to ∇. In fact, as already mentioned in the Introduction, according to Theorem 1.2 of [8] the weaker condition (∇ X Ω)(X, ·) = 0, for any vector field X, implies that (M, Q, g) is quaternionic-Kähler.

Proof of the main result
In this Section we prove our main result. Let (M, Q, g) be an almostquaternionic Hermitian manifold, whose fundamental 4-form Ω is conformal-Killing. In order to prove that Ω is parallel with respect to the Levi-Civita connection ∇, it is enough to show that it is co-closed (being conformal-Killing, Ω is co-closed if and only if it is Killing, if and only if it is parallel, by Theorem 1.2 of [8] already mentioned before). Recall now that ∇Ω is a section of T * M ⊗ (S 2 HΛ 2 0 E), which decomposes into irreducible sub-bundles as (10) Decomposition (10) follows from (7), together with the irreducible decomposition (10), they are not irreducible sub-bundles of Λ 3 (T C M), see (6). These observations readily imply that if ∇Ω is a section of HΛ 3 0 E ⊕(S 3 H)K, then Ω is co-closed: just write δΩ = − i (∇ E i Ω)(E i , ·), where {E i } is a local orthonormal frame of T M, and use the fact that an invariant linear map between non-isomorphic irreducible representations is identically zero. (Actually, by Theorem 2.3 of [8], also the converse is true: if δΩ = 0 then ∇Ω is a section of HΛ 3 0 E ⊕ (S 3 H)K).
Therefore, we aim to show that ∇Ω is a section of HΛ 3 0 E ⊕ (S 3 H)K. For this, we define the algebraic conformal-Killing operator where γ ∈ T * M (is identified with a vector using the Riemannian metric), α ∈ Λ 4 (T M) and X ∈ T M. Note that, for any 4-form ψ ∈ Ω 4 (M), In particular, since Ω is conformal-Killing, The operator T is Sp(n)Sp(1)-invariant and we extend it, by complex linearity, to T * For any irreducible sub-bundle W of S, we will determine an Sp(n)Sp(1)invariant linear map T W : T * C M ⊗ Λ 4 (T C M) → W which factors through T (i.e. T W = pr W • T is the composition of T with an Sp(n)Sp(1)-invariant linear map pr W from T * C M ⊗ Λ 4 (T C M) to W ) such that the restriction of T W to T * C M ⊗ (S 2 HΛ 2 0 E) is non-zero. An easy argument which uses (13), Schur's Lemma and the fact that irreducible subbundles of T * C M ⊗ (S 2 HΛ 2 0 E) are pairwise non-isomorphic, would then imply that ∇Ω has trivial component on W and therefore that ∇Ω is a section of HΛ 3 0 E ⊕ (S 3 H)K, as needed.
In order to define the maps T W , we apply several suitable contractions to the algebraic conformal-Killing operator T . We first define T HE and T HK as follows.
where in (15) T (η)(X) belongs to Λ 4 (T C M) (is the value of the Λ 4 (T C M)valued 1-form T (η) on X ∈ T C M) and denotes the contraction with ω E , which on decomposable multi-vectors we finally obtain T HE (η) and T HK (η), as follows.
The contraction of ω 2 Similarly, we can project ω 2 H • ω E • T (η) to H ⊗ EΛ 2 0 E and then to HK, by means of the decomposition (7) (translated to vector bundles). The result of this projection is the value of T HK on η. More precisely, Proposition 3. The operators T HE and T HK defined by (17) and (18) are non-trivial on T * C M ⊗ (S 2 HΛ 2 0 E).
In order to prove Proposition 3, we will show that T HE and T HK take non-zero value on γ 0 α 0 , where was already considered in [8]. In (19) {e 1 , · · · , e 2n } is a unitary basis of (local) sections of E, with respect to the (positive definite) Hermitian metric g E := ω E (·, j E ·), chosen such that e n+j =ẽ j for any 1 ≤ j ≤ n, and {h,h} is a unitary basis of (local) sections of H, with respect to g H := ω H (·, j H ·). In order to simplify notations, in (19) and bellow we omit the summation sign over 1 ≤ i ≤ 2n. The symplectic forms of E and H can be written as From (9) and (20), α 0 is a section of the sub-bundle S 2 HΛ 2 0 E of Λ 4 (T C M) and γ 0 α 0 is a section of T * C M ⊗ (S 2 HΛ 2 0 E).
We divide the proof of Proposition 3 into the following two Lemmas.

Proof. A straightforward computation shows that
Therefore, using (11), we can write where γ 0 ∧ α 0 (·) is a 1-form with values in Λ 4 (T C M), whose natural contraction with a vector X ∈ T C M is γ 0 ∧ i X α 0 . Similarly, F and G are defined by Now, it is straightforward to check that These relations combined with (21) readily imply that with constants and Projecting the expression for ω 2 which is not totally skew in the E-variables. Our claim follows.
Lemma 5. The value of T HE on γ 0 α 0 is In particular, T HE (γ 0 α 0 ) is non-zero.
Proof. The claim follows from a straightforward calculation, using the ex- We now define the maps T (S 3 H)E and T S 3 HΛ 3 0 E . For a section η of T * C M ⊗ Λ 4 (T C M), T (η) is a section of EH ⊗ Λ 4 (T C M). We consider ω H • T (η), the contraction of T (η) with ω H in the first two H-variables, which is a section of EE ⊗ Λ 3 (T C M). Its total symmetrization sym H (ω H • T (η)) in the Hvariables is a section of EE(S 3 H)Λ 3 E. Leaving the first two E-variables of sym H (ω H • T (η)) unchanged and contracting sym H (ω H • T (η)) with ω E on Λ 3 E, as in (2), we get a section ω E • sym H (ω H • T (η)) of EE(S 3 H)E.
To define T (S 3 H)Λ 3 0 E (η) and T (S 3 H)E (η) we project ω E • sym H (ω H • T (η)) on (S 3 H)Λ 3 E and then we project the result on (S 3 H)Λ 3 0 E and (S 3 H)E respectively, using the decomposition (3), with r = 3. Therefore, Similarly, is the contraction of pr ( Proposition 6. The operators T (S 3 H)Λ 3 0 E and T (S 3 H)E defined by (23) and (24) are non-trivial on T * C M ⊗ (S 2 HΛ 2 0 E). Like in the proof of Proposition 3, we will show that T (S 3 H)Λ 3 0 E (γ 0 α 0 ) and T (S 3 H)E (γ 0 α 0 ) are non-zero. This is a consequence of the next Lemma.
Corollary 8 implies Proposition 6. Proposition 3 and Proposition 6 conclude the proof of our main result.

Acknowledgements
I grateful to Paul Gauduchon for many useful discussions about conformal-Killing forms and to Uwe Semmelmann for his interest in this work. This work was supported by Consiliul National al Cercetarii Stiintifice din Invatamantul Superior, through a CNCSIS grant IDEI "Structuri geometrice pe varietati diferentiabile", [code 1187/2008].