The twistor spaces of a para-quaternionic Kähler manifold

Dmitri V. Alekseevsky

Dept. of Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK D.V.Alekseevsky@maths.hull.ac.uk

Vicente Cortés

Institut Élie Cartan de Mathématiques, Université Henri Poincaré - Nancy 1, B.P. 239,

F-54506 Vandœuvre-lès-Nancy Cedex, France cortes@iecn.u-nancy.fr

December 19, 2005

Abstract

We develop the twistor theory of G-structures for which the (linear) Lie algebra of the structure group contains an involution, instead of a complex structure. The twistor space Z of such a G-structure is endowed with a field of involutions 𝒥 ∈ Γ(End T Z) and a 𝒥-invariant distribution ℋ_{Z}. We study the conditions for the integrability of 𝒥 and for the (para-)holomorphicity of ℋ_{Z}. Then we apply this theory to para-quaternionic Kähler manifolds of non-zero scalar curvature, which admit two natural twistor spaces (Z^{ϵ}, 𝒥, ℋ_{Z}), ϵ = ±1, such that 𝒥^{2} = ϵ Id. We prove that in both cases 𝒥 is integrable (recovering results of Blair, Davidov and Mus̆karov) and that ℋ_{Z} defines a holomorphic (ϵ = −1) or para-holomorphic (ϵ = +1) contact structure. Furthermore, we determine all the solutions of the Einstein equation for the canonical one-parameter family of pseudo-Riemannian metrics on Z^{ϵ}. In particular, we find that there is a unique Kähler-Einstein (ϵ = −1) or para-Kähler-Einstein (ϵ = +1) metric. Finally, we prove that any Kähler or para-Kähler submanifold of a para-quaternionic Kähler manifold is minimal and describe all such submanifolds in terms of complex (ϵ = −1), respectively, para-complex (ϵ = +1) submanifolds of Z^{ϵ} tangent to the contact distribution.

Contents

1 Introduction 2

2 (Almost) para-complex manifolds 3 2.1 Integrability of an almost para-complex structure . . . . . . . . . . . . . . 3 2.2 Holomorphicity of distributions in almost para-complex manifolds . . . . . 5

3 Para-quaternionic manifolds and para-quaternionic Kähler manifolds 7

4 The twistor spaces of a para-quaternionic or para-quaternionic Kähler manifold 9 4.1 The twistor spaces of a para-quaternionic manifold . . . . . . . . . . . . . 9 4.2 The twistor spaces of a para-quaternionic Kähler manifold . . . . . . . . . 10 4.3 Twistor spaces of para-quaternionic (Kähler) manifolds as bundles associ-

ated to G-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 G-structures of para-twistor type and their twistor spaces: obstructions for integrability 12 5.1 Groups of para-twistor type and para-complex symmetric spaces . . . . . . 12 5.2 The space of curvature tensors . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.3 G-structures with connection and associated K-structures . . . . . . . . . . 15 5.4 The twistor space of a G-structure of para-twistor type . . . . . . . . . . . 17

6 Integrability and holomorphicity results for the twistor spaces of a paraquaternionic Kähler manifold 20

7 The canonical ϵ-Kähler-Einstein metric and contact structure on the twistor space Z^{ϵ} of a para-quaternionic Kähler manifold 21

8 Twistor construction of minimal submanifolds of para-quaternionic Kähler manifolds 30 8.1 Kähler and para-Kähler submanifolds of para-quaternionic Kähler manifolds 30 8.2 Twistor construction of Kähler and para-Kähler submanifolds of para-

quaternionic Kähler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 32

1 Introduction

Twistor methods were originally introduced by Penrose with the aim of providing a mathematical framework which could lead to a synthesis of quantum theory and relativity [P]. They have proven very fruitful for the construction and systematic study of various geometric objects governed by non-linear partial differential equations such as Yang-Mills connections, Einstein metrics, harmonic maps and minimal submanifolds.

Given a geometric problem on a real differentiable manifold M endowed with certain geometric structure S, the twistor approach is to try to translate the given problem into a problem of complex geometry an a complex manifold Z, called the twistor space, which is the total space of a bundle over M. In most cases, Z can be defined as the bundle of all complex structures in the tangent spaces of M which are compatible with the geometric structure S and it comes with a natural almost complex structure 𝒥, the integrability of which has to be derived from the properties of the structure S.

In the case of a four-dimensional oriented Riemannian manifold M, for instance, the fibre at p ∈ M of the twistor bundle Z → M consists of all skew-symmetric complex structures in T_{p}M, which induce the given orientation [AHS]. It is identified with the Riemann sphere ℂP^{1} and, thus, carries a natural complex structure. On the other hand, the Levi-Civita connection of M induces a horizontal (i.e. transversal to the fibers) distribution ℋ_{Z} ⊂ T Z and the horizontal spaces carry a tautological complex structure. Putting the complex structures on vertical and horizontal spaces together, one obtains a canonical almost complex structure 𝒥 on Z. By the results of Atiyah, Hitchin and Singer, 𝒥 is integrable if and only if the Weyl curvature tensor of M is self-dual and, in that case, self-dual Yang-Mills vector bundles on M correspond to certain holomorphic vector bundles on Z. Salamon et al. have extended these constructions from four to higher dimensions, with the role of the self-dual four-dimensional Riemannian manifold played by a quaternionic Kähler manifold [S, CS]. In [AM] the twistor method was used to construct (minimal) Kähler submanifolds of quaternionic Kähler manifolds.

A G-structure is called of twistor type if the (linear) Lie algebra 𝔤 = Lie G of the structure group contains a complex structure, i.e. an element J such that J^{2} = −Id. The twistor theory of G-structures of twistor type is developped in [AG], see also references therein. This includes the case of quaternionic Kähler manifolds, for which the structure group is G = Sp(1)Sp(n).

In this paper, we develop a similar theory for G-structures of para-twistor type, i.e. for which 𝔤 contains an involution J, rather than a complex structure. Let P → M be such a G-structure and denote by K = Z_{G}(J) the centralizer of the involution J. For any principal connection ω on P, we define the twistor space of (P, ω) as the total space of the bundle Z = P/K → P/G = M, which we endow with a K-structure P → Z, a field of involutions 𝒥 ∈ Γ(End T Z) and a 𝒥-invariant horizontal distribution ℋ_{Z}, see Def.

11. We express the integrability of 𝒥 and the (para-)holomorphicity of ℋ_{Z} as equations for the curvature and torsion of ω, which generalize the self-duality equation for the Weyl curvature of a pseudo-Riemannian metric of signature (2, 2), see Thm. 1.

A para-quaternionic structure on a vector space V is a Lie subalgebra Q ⊂ End V which admits a basis (J_{1}, J_{2}, J_{3}) such that J_{3} = J_{1}J_{2} and J_{α}^{2} = ϵ_{α}Id, where (ϵ_{1}, ϵ_{2}, ϵ_{3}) = (−1, 1, 1). A pseudo-Riemannian manifold (M, g) of dimension > 4 endowed with a parallel field M ∋ p ↦ Q_{p} ⊂ End T_{p}M of g-skew-symmetric para-quaternionic structures is called a para-quaternionic Kähler manifold. The metric g has signature (2n, 2n) and is Einstein [AC]. Moreover, para-quaternionic Kähler manifolds are related to certain supersymmetric field theories on space-times with a positive definite rather than a Lorentzian metric [CMMS2].

For a para-quaternionic Kähler manifold (M, g, Q), Blair et al. [BDM, B] have defined two twistor spaces Z^{ϵ} := {A ∈ Q|A^{2} = ϵ}, ϵ = ±1, and endowed them with an integrable structure 𝒥 ⊂ End T Z^{ϵ} such that 𝒥^{2} = ϵId. We recover these results by considering the twistor space associated to the underlying G-structure, which is of twistor type, as well as of para-twistor type. More precisely, we consider

J ∈ 𝔰𝔩(2, ℝ) ⊂ 𝔤 = 𝔰𝔩(2, ℝ) ⊕ 𝔰𝔭(ℝ^{2n}) ⊂ 𝔤𝔩(ℝ^{2} ⊗ ℝ^{2n}) = 𝔤𝔩(4n, ℝ) .

Under the assumption that the scalar curvature of g is non-zero, we prove, in addition, that the horizontal distribution ℋ_{Z} defines a holomorphic (respectively, para-holomorphic) contact structure on Z and that (Z^{ϵ}, 𝒥) admits a Kähler-Einstein (respectively, para-Kähler-Einstein) metric and determine all Einstein metrics in the canonical one-parameter family of pseudo-Riemannian metrics, see Thm. (3). It turns out that there is always a second Einstein metric.

Finally, we generalize the twistor construction of Kähler submanifolds of a quaternionic Kähler manifold (see [AM]) to the case of Kähler and para-Kähler submanifolds (see Def.

14) of a para-quaternionic Kähler manifold (M, g, Q). We prove that any Kähler or para-Kähler submanifold of a para-quaternionic Kähler manifold (M, g, Q) is minimal (Cor. 7). All such submanifolds can be obtained as projections of complex (ϵ = −1), respectively, para-complex (ϵ = +1) submanifolds of Z^{ϵ} which are tangent to the contact distribution, see Thm. 4. It follows that the maximal dimension of a Kähler or para-Kähler submanifold of (M, g, Q) is \frac{1}{2} dim M and that maximal Kähler (respectively, para-Kähler) submanifolds of (M, g, Q) correspond to Legendrian submanifolds of the complex (respectively, paracomplex) contact manifold (Z^{ϵ}, ℋ_{Z}).

2 (Almost) para-complex manifolds

2.1 Integrability of an almost para-complex structure

Definition 1 An (almost) para-complex structure, in the weak sense, on a differentiable manifold M is a field of endomorphisms J ∈ End T M such that J^{2} = Id. J is called non-trivial if J ≠ ±Id. We say that J is an (almost) para-complex structure, in the strong sense, if the ±1-eigenspace distributions T^{±}M of J have the same rank. An almost paracomplex structure is called integrable, or para-complex structure if the distributions T^{±}M are integrable, or, equivalently, the Nijenhuis tensor N_{J}, defined by

N_{J}(X, Y) = [X, Y] + [JX, JY] − J[JX, Y] − J[X, JY] , X, Y ∈ T M ,  (2.1)

vanishes. An (almost) para-complex manifold (M, J) is a manifold M endowed with an (almost) para-complex structure.

Unless otherwise stated, by an (almost) para-complex structure we shall understand here an (almost) para-complex structure in the weak sense.

Remark: The difference between weak and strong (almost) para-complex manifolds is that T_{p}^{1,0}M = {X + eJX|X ∈ T_{p}M} ⊂ T_{p}M ⊗ C, p ∈ M, is a free module over the ring C := ℝ[e], e^{2} = 1, of para-complex numbers only in the strong case. In particular, for weak para-complex manifolds, there is no notion of para-holomorphic local coordinates (z^{i}) on M such that the (dz^{i}) form a basis of T_{p}^{1,0}M over C.

Let (V, J) and (U, J_{U}) be vector spaces endowed with constant para-complex structures. We can decompose the vector space C^{2}(U) := U ⊗ ∧^{2}V^{∗} of U-valued two-forms on V according to type

C^{2}(U) = ∑_{p+q=2} C^{p,q}(U) ,  (2.2)

where

α ∈ C^{1,1}(U) if α(JX, JY) = −α(X, Y) for all X, Y ∈ V ,
α ∈ C^{2,0}(U) if α(JX, Y) = α(X, JY) = J_{U}α(X, Y) for all X, Y ∈ V and
α ∈ C^{0,2}(U) if α(JX, Y) = α(X, JY) = −J_{U}α(X, Y) for all X, Y ∈ V .

Lemma 1 The projections π^{p,q} : C^{2}(U) → C^{p,q}(U), α → α^{p,q}, are given by:

α^{1,1}(X, Y) = \frac{1}{2}(α(X, Y) − α(JX, JY))
α^{2,0}(X, Y) = \frac{1}{4}(α(X, Y) + α(JX, JY) + J_{U}α(JX, Y) + J_{U}α(X, JY))
α^{0,2}(X, Y) = \frac{1}{4}(α(X, Y) + α(JX, JY) − J_{U}α(JX, Y) − J_{U}α(X, JY))

For scalar valued forms (U = ℝ) we will always assume that J_{U} = Id.

Let J be an almost para-complex structure on a manifold M and ∇ a linear connection which preserves J. The following lemma shows that J is integrable if and only if the (0, 2) component T^{0,2} = π^{0,2}T vanishes.

Proposition 1 Let ∇ be a connection which preserves an almost para-complex structure J on a manifold M. Then the Nijenhuis tensor of J is given by N_{J} = −4T^{0,2}. In particular, J is integrable if and only if T^{0,2} = 0.

Proof: Applying Lemma 1 in the case U = V = T_{p}M, p ∈ M, we have

T^{0,2}(X, Y) = \frac{1}{4}(T (X, Y) + T (JX, JY) − JT (JX, Y) − JT (X, JY)) , X, Y ∈ T M .

Replacing T (X, Y) by ∇_{X}Y − ∇_{Y} X − [X, Y] in this formula, we get

T^{0,2}(X, Y) = −\frac{1}{4}([X, Y] + [JX, JY] − J[JX, Y] − J[X, JY]) = −\frac{1}{4}N_{J}(X, Y) .

2.2 Holomorphicity of distributions in almost para-complex manifolds

Definition 2 Let (M, J) be an almost para-complex manifold of real dimension n. A J-invariant distribution 𝒟 = 𝒟_{+} ⊕ 𝒟_{−} ⊂ T^{+}M ⊕ T^{−}M = T M of rank m is called paraholomorphic if it is locally defined by equations α_{+}^{1} = ⋯ = α_{+}^{k_{+}} = α_{−}^{1} = ⋯ = α_{−}^{k_{−}} = 0, such that k_{+} + k_{−} = n − m,

α_{±}^{i} ∘ J = ±α_{±}^{i}  (2.3)

and the (1, 1)-component

π^{1,1}dα_{+}^{i} = \frac{1}{2}(dα_{+}^{i} − J^{∗}dα_{+}^{i})

vanishes on ∧^{2}(𝒟_{+} ⊕ T^{−}M) and the (1, 1)-component

π^{1,1}dα_{−}^{i} = \frac{1}{2}(dα_{−}^{i} − J^{∗}dα_{−}^{i})

vanishes on ∧^{2}(T^{+}M ⊕ 𝒟_{−}).

Let (M, J) be an almost para-complex manifold of real dimension n endowed with a J-invariant distribution 𝒟 ⊂ T M of rank m and a connection ∇ which preserves J and 𝒟. Then we can define a two-form with values in T M/𝒟 by

S(X, Y) := T (X, Y) mod 𝒟 .

Since J induces a para-complex structure on the vector bundle T M/𝒟, we can decompose

S = S^{2,0} + S^{1,1} + S^{0,2} ,

see Lemma 1.

Proposition 2 Let (M, J) be an almost para-complex manifold. A J-invariant distribution 𝒟 = 𝒟_{+} ⊕ 𝒟_{−} ⊂ T^{+}M ⊕ T^{−}M = T M is para-holomorphic if and only if

[Γ(𝒟_{±}), Γ(T^{∓}M)] ⊂ Γ(T^{∓}M ⊕ 𝒟_{±}) .  (2.4)

Moreover, if ∇ is a connection which preserves J and 𝒟, then (2.4) is equivalent to

S^{1,1}(JX, ·) = −JS^{1,1}(X, ·) ,  (2.5)

for all X ∈ 𝒟.

Proof: First we prove that (2.5) is equivalent to the para-holomorphicity of 𝒟. Let 𝒟 be a para-holomorphic distribution defined by one-forms α_{±}^{i} as in Definition 2. The condition on π^{1,1}dα_{±}^{i} is equivalent to

dα_{+}^{i}(X_{+}, Y_{−}) = 0 , X_{+} ∈ 𝒟_{+} , Y_{−} ∈ T^{−}M ,
dα_{−}^{i}(X_{+}, Y_{−}) = 0 , X_{+} ∈ T^{+}M , Y_{−} ∈ 𝒟_{−} .

Expressing the exterior derivative in terms of the covariant derivative and torsion we get

0 = dα_{+}^{i}(X_{+}, Y_{−}) = (∇_{X_{+}}α_{+}^{i})Y_{−} − (∇_{Y_{−}}α_{+}^{i})X_{+} + α_{+}^{i}(T (X_{+}, Y_{−})) .

The first two terms on the right-hand side vanish. In fact, since ∇ preserves the distribution 𝒟, the covariant derivative ∇_{X}α_{+}^{i} vanishes on 𝒟_{+} ⊕ T^{−}M for all X ∈ T M. The last term can be written as

0 = α_{+}^{i}(T (X_{+}, Y_{−})) = α_{+}^{i}(T^{1,1}(X_{+}, Y_{−})) ,

which implies that T^{1,1}(X_{+}, Y_{−}) ∈ 𝒟_{+} + T^{−}M for all X_{+} ∈ 𝒟_{+} and Y_{−} ∈ T^{−}M. A similar calculation for α_{−}^{i} shows that T^{1,1}(X_{+}, Y_{−}) ∈ T^{+}M ⊕ 𝒟_{−} for all X_{+} ∈ T^{+}M and Y_{−} ∈ 𝒟_{−}. This proves that

S^{1,1}(𝒟_{+}, T^{−}M) ⊂ (T^{−}M + 𝒟)/𝒟
S^{1,1}(𝒟_{−}, T^{+}M) ⊂ (T^{+}M + 𝒟)/𝒟 .

In particular, S^{1,1}(𝒟, 𝒟) = 0 and S^{1,1}(JX, ·) = −JS^{1,1}(X, ·) for all X ∈ 𝒟.

To prove the converse, we assume that the torsion of ∇ satisfies (2.5). Let (α_{+}^{1}, ⋯, α_{+}^{k_{+}}) and (α_{−}^{1}, ⋯, α_{−}^{k_{−}}) be local frames of (𝒟_{+} ⊕ T^{−}M)^{⊥} and (T^{+}M ⊕ 𝒟_{−})^{⊥} ⊂ T^{∗}M, respectively. This implies (2.3). Since π^{1,1}α(T^{±}M, T^{±}M) = 0 for any two-form α, it is sufficient to check that π^{1,1}dα_{+}^{i}(𝒟_{+}, T^{−}M) = π^{1,1}dα_{−}^{i}(T^{+}M, 𝒟_{−}) = 0. We calculate for X_{+} ∈ 𝒟_{+} and Y_{−} ∈ T^{−}M:

π^{1,1}dα_{+}^{i}(X_{+}, Y_{−}) = dα_{+}^{i}(X_{+}, Y_{−}) = (∇_{X_{+}}α_{+}^{i})Y_{−} − (∇_{Y_{−}}α_{+}^{i})X_{+} + α_{+}^{i}(T (X_{+}, Y_{−}))
= α_{+}^{i}(T (X_{+}, Y_{−})) = α_{+}^{i}(T^{1,1}(X_{+}, Y_{−})) = α_{+}^{i}(S^{1,1}(X_{+}, Y_{−}))
= α_{+}^{i}(S^{1,1}(JX_{+}, Y_{−})) =^{(2.5)} −α_{+}^{i}(JS^{1,1}(X_{+}, Y_{−})) = −α_{+}^{i}(S^{1,1}(X_{+}, Y_{−})).

Therefore, π^{1,1}dα_{+}^{i}(X_{+}, Y_{−}) = 0. A similar calculation shows that π^{1,1}dα_{−}^{i}(𝒟_{−}, T^{+}M) = 0.

Now we prove the equivalence of (2.4) and (2.5). The condition (2.5) can be written as

T (𝒟_{±}, T^{∓}M) ⊂ T^{∓}M ⊕ 𝒟_{±} .

Using that ∇ preserves the distributions 𝒟_{±} and T^{±}M, we calculate for X_{±} ∈ Γ(𝒟_{±}) and Y_{±} ∈ Γ(T^{±}M)

T^{∓}M ⊕𝒟_{±} ∋ T (X_{±}, Y_{∓}) = ∇_{X_{±}}Y_{∓}−∇_{Y_{∓}}X_{±}−[X_{±}, Y_{∓}] ≡ −[X_{±}, Y_{∓}] mod T^{∓}M ⊕𝒟_{±} .

This proves the equivalence of (2.4) and (2.5).

Let (M, J) be a para-complex manifold in the strong sense, i.e. the integrable eigendistributions T^{±}M are of the same rank. Recall [CMMS1] that a C-valued one-form γ = α + eβ is of para-complex type (1, 0), i.e. J^{∗}γ = eγ, if and only if β = α ∘ J. A (1, 0)-form γ is para-holomorphic if ¯∂γ := π^{1,1}dγ = 0, which is equivalent to the para-Cauchy-Riemann equations

∂_{−}α_{+} := π^{1,1}dα_{+} = ∂_{+}α_{−} := π^{1,1}dα_{−} = 0 ,  (2.6)

where α = α_{+} + α_{−} is the J-eigenspace decomposition of α.

Proposition 3 Let (M, J) be a para-complex manifold in the strong sense with eigendistributions T^{±}M of rank n and 𝒟 = 𝒟_{+} ⊕ 𝒟_{−} ⊂ T^{+}M ⊕ T^{−}M = T M a J-invariant distribution such that 𝒟_{±} are of the same rank m. Then 𝒟 is para-holomorphic if and only if it is locally defined by equations γ^{i} = 0 (i = 1, ⋯, k = n − m), where the γ^{i} are para-holomorphic one-forms.

Proof: Let 𝒟 be defined by para-holomorphic one-forms γ^{i} = α_{+}^{i} + α_{−}^{i} + e(α_{+}^{i} − α_{−}^{i}). The α_{±}^{i} satisfy (2.6), which imply the equations in the Definition 2.

To prove the converse, we now asume that the distribution 𝒟 is para-holomorphic. Thanks to Proposition 2, this means that

[Γ(𝒟_{±}), Γ(T^{∓}M)] ⊂ Γ(T^{∓}M ⊕ 𝒟_{±}) .

In order to construct para-holomorphic one-forms γ^{i} = α_{+}^{i} + α_{−}^{i} + e(α_{+}^{i} − α_{−}^{i}) which define 𝒟, we choose locally linearly independent commuting vector fields Y_{i}^{±} ∈ Γ(T^{±}M) which generate distributions N^{±} ⊂ T^{±}M complementary to 𝒟_{±}. We define one-forms α_{±}^{i} vanishing on 𝒟_{±} ⊕ T^{∓}M by

α_{±}^{i}(Y_{j}^{±}) = δ_{j}^{i} .

It is clear that α_{±}^{i} ∘ J = ±α_{±}^{i} and that γ^{i} := α_{+}^{i} + α_{−}^{i} + e(α_{+}^{i} − α_{−}^{i}) define 𝒟. Now we check that the γ^{i} are para-holomorphic, i.e. ∂_{−}α_{+}^{i} = ∂_{+}α_{−}^{i} = 0. It is sufficient to evaluate this equality on (Z^{+}, Z^{−}), where Z^{±} = X^{±} ∈ Γ(𝒟_{±}) or Z^{±} = Y_{i}^{±}.

∂_{−}α_{+}^{i}(X^{+}, X^{−}) = X^{+}α_{+}^{i}(X^{−}) − X^{−}α_{+}^{i}(X^{+}) − α_{+}^{i}([X^{+}, X^{−}]) = 0 ,

since α_{+}^{i} vanishes on 𝒟_{+} ⊕ T^{−}M and [X^{+}, X^{−}] ∈ T^{−}M ⊕ 𝒟_{+} by (2.4). Similarly,

∂_{−}α_{+}^{i}(X^{+}, Y_{j}^{−}) = X^{+}α_{+}^{i}(Y_{j}^{−}) − Y_{j}^{−}α_{+}^{i}(X^{+}) − α_{+}^{i}([X^{+}, Y_{j}^{−}]) = 0 .

Finally,

∂_{−}α_{+}^{i}(Y_{j}^{+}, Y_{k}^{−}) = Y_{j}^{+}α_{+}^{i}(Y_{k}^{−}) − Y_{k}^{−}α_{+}^{i}(Y_{j}^{+}) − α_{+}^{i}([Y_{j}^{+}, Y_{k}^{−}]) = 0 − Y_{k}^{−}(δ_{j}^{i}) − 0 = 0 ,

since, by construction, [Y_{j}^{+}, Y_{k}^{−}] = 0. Similarly, one can check that ∂_{+}α_{−}^{i} = 0.

3 Para-quaternionic manifolds and para-quaternionic Kähler manifolds

Definition 3 Let (ϵ_{1}, ϵ_{2}, ϵ_{3}) = (−1, 1, 1), or a permutation thereof. An almost paraquaternionic structure on a differentiable manifold M (of dimension 4n) is a rank 3 subbundle Q ⊂ End T M, which is locally generated by three anticommuting fields of endomorphisms J_{1}, J_{2}, J_{3} = J_{1}J_{2}, such that J_{α}^{2} = ϵ_{α}Id. Such a triple (J_{α}) will be called a standard local basis of Q. A linear connection which preserves Q is called an almost para-quaternionic connection. An almost para-quaternionic structure Q is called a para-quaternionic structure if M admits a para-quaternionic connection, i.e. a torsion-free connection which preserves Q. An (almost) para-quaternionic manifold is a manifold endowed with an (almost) paraquaternionic structure.

An almost para-quaternionic Hermitian manifold (M, g, Q) is a pseudo-Riemannian manifold (M, g) endowed with a para-quaternionic structure Q consisting of skew-symmetric endomorphisms. (M, g, Q), n > 1, is called a para-quaternionic Kähler manifold if the Levi-Civita connection preserves Q.

Proposition 4 [AC] At any point, the curvature tensor R of a para-quaternionic Kähler manifold (M, g, Q) of dimension 4n > 4 admits a decomposition

R = νR_{0} + W ,  (3.1)

where ν = \frac{scal}{4n(n+2)} is the reduced scalar curvature,

R_{0}(X, Y) := +\frac{1}{2} ∑_{α} ϵ_{α}g(J_{α}X, Y)J_{α} + \frac{1}{4} (X ∧ Y − ∑_{α} ϵ_{α}J_{α}X ∧ J_{α}Y ) , X, Y ∈ T_{p}M ,

is the curvature tensor of the para-quaternionic projective space of the same dimension as M and W is a trace-free Q-invariant algebraic curvature tensor, where Q acts by derivations. In particular, R is Q-invariant.

We define a para-quaternionic Kähler manifold of dimension 4 as a pseudo-Riemannian manifold endowed with a parallel skew-symmetric para-quaternionic structure whose curvature tensor admits a decomposition (3.1).

Since the Levi-Civita connection ∇ of a para-quaternionic Kähler manifold preserves the para-quaternionic structure Q, we can write

∇J_{α} = −ϵ_{β}ω_{γ} ⊗ J_{β} + ϵ_{γ}ω_{β} ⊗ J_{γ} ,  (3.2)

where (α, β, γ) is a cyclic permutation of (1, 2, 3). We shall denote by ρ_{α} := g(J_{α}·, ·) the fundamental form associated with J_{α} and put ρ′_{α} := −ϵ_{α}ρ_{α}.

Proposition 5 The locally defined fundamental forms satisfy the following structure equations

νρ′_{α} := −ϵ_{α}νρ_{α} = ϵ_{3}(dω_{α} − ϵ_{α}ω_{β} ∧ ω_{γ}) ,  (3.3)

where (α, β, γ) is a cyclic permutation of (1, 2, 3).

Proof: Using Proposition 4 and the fact that

[J_{α}, J_{β}] = 2ϵ_{3}ϵ_{γ}J_{γ} ,

we calculate the action of the curvature operator R(X, Y), X, Y ∈ T M, on J_{α}:

[R(X, Y), J_{α}] = [νR_{0}(X, Y), J_{α}] = −\frac{ν}{2} ∑_{δ=1}^{3} ρ′_{δ}(X, Y)[J_{δ}, J_{α}]
= ϵ_{3}ν(−ϵ_{β}ρ′_{γ}(X, Y)J_{β} + ϵ_{γ}ρ′_{β}(X, Y)J_{γ}) ,

where (α, β, γ) is a cyclic permutation of (1, 2, 3). On the other hand, using the equation (3.2), we calculate

[R(X, Y), J_{α}] = [∇_{X}, ∇_{Y}]J_{α} − ∇_{[X,Y]}J_{α}
= ∇_{X}(−ϵ_{β}ω_{γ}(Y)J_{β} + ϵ_{γ}ω_{β}(Y)J_{γ}) − ∇_{Y} (−ϵ_{β}ω_{γ}(X)J_{β} + ϵ_{γ}ω_{β}(X)J_{γ})
−(−ϵ_{β}ω_{γ}([X, Y])J_{β} + ϵ_{γ}ω_{β}([X, Y])J_{γ})
= −ϵ_{β}dω_{γ}(X, Y)J_{β} + ϵ_{γ}dω_{β}(X, Y)J_{γ} − ϵ_{β}ω_{γ}(Y)∇_{X}J_{β} + ϵ_{γ}ω_{β}(Y)∇_{X}J_{γ}
+ϵ_{β}ω_{γ}(X)∇_{Y} J_{β} − ϵ_{γ}ω_{β}(X)∇_{Y} J_{γ} .

Applying again the equation (3.2), we finally get

[R(X, Y), J_{α}] = −ϵ_{β}(dω_{γ} − ϵ_{γ}ω_{α} ∧ ω_{β})(X, Y)J_{β} + ϵ_{γ}(dω_{β} − ϵ_{β}ω_{γ} ∧ ω_{α})(X, Y)J_{γ} .

Comparing the two formulas for [R(X, Y), J_{α}] we obtain the structure equations.

4 The twistor spaces of a para-quaternionic or paraquaternionic Kähler manifold

4.1 The twistor spaces of a para-quaternionic manifold

In the following, it will be useful to unify complex and para-complex structures in the following definition.

Definition 4 An almost ϵ-complex structure, ϵ ∈ {−1, 0, 1}, on a differentiable manifold M of dimension 2n is a field of endomorphisms J ∈ End T M such that J^{2} = ϵId and, moreover, for ϵ = +1 the eigendistributions T^{±}M are of rank n and for ϵ = 0 the two distributions ker J and im J have rank n. In other words, an almost −1-complex structure is an almost complex structure and an almost +1-complex structure is an almost paracomplex structure in the strong sense. An ϵ-complex manifold is a differentiable manifold endowed with an integrable (i.e. N_{J} = 0) ϵ-complex structure J.

We shall also use the unifying adjective ϵ-holomorphic as a synonym of ‘holomorphic’ or ‘para-holomorphic’, depending on whether ϵ = −1 or ϵ = +1, respectively.

Let (M, Q) be an almost para-quaternionic manifold. We associate with (M, Q) a family of bundles π : Z^{s} → M, with two-dimensional fibres, depending on a parameter s ∈ ℝ as follows:

Z^{s} := {A ∈ Q|A ≠ 0 , A^{2} = s} .

Definition 5 The fibre bundle π : Z^{s} → M is called the s-twistor space of the almost para-quaternionic manifold (M, Q).

Proposition 6 Any almost para-quaternionic connection ∇ on an almost para-quater-nionic manifold (M, Q) induces a canonical almost ϵ-complex structure 𝒥^{s} = 𝒥_{∇}^{s} on the s-twistor space Z^{s}, where ϵ = sgn(s) ∈ {−1, 0, 1}.

Proof: Let (I, J, K) be a standard basis of Q_{m}. Then any element A ∈ Q_{m} can be written as A = xI + yJ + zK and A ∈ Z^{s} if and only if −x^{2} + y^{2} + z^{2} = s. Hence, the fibres of Z^{s} are two-sheeted hyperboloids for s < 0, one-sheeted hyperboloids for s > 0 and light-cones without origin for s = 0. Each fibre Z_{m}^{s} = π^{−1}(m) is a homogeneous space of the group SO(1, 2) with one-dimensional stabilizer SO(1, 2)_{A_{s}} = SO(2) if s < 0, SO(1, 2)_{A_{s}} = SO(1, 1) if s > 0 and SO(1, 2)_{A_{s}} ≅ (ℝ, +) if s = 0, where A_{s} ∈ Z^{s}. First we define the canonical SO(1, 2)-invariant ϵ-complex structure on Z_{m}^{s}, as follows. The threedimensional vector space Q_{m} ⊂ End T_{m}M is a Lie subalgebra isomorphic to 𝔰𝔩_{2}(ℝ). The adjoint action preserves the indefinite scalar product ⟨A, B⟩ = −\frac{1}{4n}tr(AB), 4n = dim M, in Q and hence identifies the Lie algebra Q with 𝔰𝔬(Q) = Lie SO(Q) ≅ 𝔰𝔬(1, 2). Let A ∈ Z_{m}^{s} ⊂ Q_{m}. Then Z_{m}^{s} = SO(Q)A and the tangent space to Z_{m}^{s} at A is identified with 𝔰𝔬(Q)A ≅ 𝔰𝔬(Q)/𝔰𝔬(Q)_{A} = 𝔰𝔬(Q)/ℝA. It is easy to check that the adjoint action of \frac{1}{2}A on 𝔰𝔬(Q)/ℝA defines an SO(Q)-invariant ϵ-complex structure J^{v} on Z_{m}^{s}. Now we define an almost ϵ-complex structure 𝒥^{s} on the twistor space Z^{s}. We have the decomposition

T_{z}Z^{s} = T_{z}^{v}Z^{s} + H_{z} ≅ T_{z}(Z_{m}^{s}) ⊕ T_{πz}M ,  (4.1)

where T_{z}^{v}Z^{s} is the vertical space of the bundle π : Z^{s} → M and H_{z} is the horizontal space of the connection in the bundle π induced by the para-quaternionic connection ∇ of (M, Q). The latter is identified with T_{πz}M via the projection Z^{s} → M. We denote by J^{z} the tautological ϵ-complex structure on T_{πz}M defined by z ∈ Z^{s}. With respect to the above decomposition we define

𝒥_{z}^{s} = J^{v} ⊕ J^{z}  (4.2)

By construction, 𝒥^{s} is an almost ϵ-complex structure.

4.2 The twistor spaces of a para-quaternionic Kähler manifold

Let (M, g, Q) be a para-quaternionic Kähler manifold with twistor spaces Z^{s}. The Levi-Civita connection ∇ = ∇^{g} is a para-quaternionic connection and, hence, induces a canonical almost ϵ-complex structure 𝒥^{s} = 𝒥_{∇}^{s} on Z^{s}.

Proposition 7 The twistor space Z^{s} of a para-quaternionic Kähler manifold (M, g, Q) admits a canonical almost ϵ-complex structure 𝒥^{ϵ}, where ϵ = sgn(s), and a one-parameter family g_{t}^{s}, t ∈ ℝ − {0}, of pseudo-Riemannian metrics such that the almost ϵ-complex structure 𝒥^{s} is skew-symmetric, provided that s ≠ 0. For s = 0 there exists a canonical one-parameter family g_{t}^{0}, t ∈ ℝ − {0}, of symmetric bilinear forms with one-dimensional (vertical) kernel such that 𝒥^{s} is skew-symmetric. Finally, for s ≠ 0, the projection π : (Z^{s}, g_{t}^{s}) → (M, g) is a pseudo-Riemannian submersion.

Proof: We denote by g^{v} := ⟨·, ·⟩|_{Z_{m}^{s}} the induced metric on the fibres Z_{m}^{s} ⊂ (Q, ⟨·, ·⟩). It is nondegenerate for s ≠ 0 and has one-dimensional kernel for s = 0. The ϵ-complex structure J^{v} on Z_{m}^{s} is g^{v}-skew-symmetric. With respect to the decomposition (4.1), we define

(g_{t}^{s})_{z} = tg^{v} ⊕ g_{πz} .

The almost ϵ-complex structure 𝒥^{s} defined above is skew-symmetric with respect to the field of symmetric bilinear forms g_{t}^{s}, which is nondegenerate for s ≠ 0 and has one-dimen-sional vertical kernel for s = 0. The above formula for g_{t}^{s} shows that the decomposition of T Z into vertical and horizontal space is g_{t}^{s}-orthogonal and that the projection induces an isometry H_{z} → T_{πz}M. This proves that π is a pseudo-Riemannian submersion.

The scalar multiplication by |s|^{1/2} ≠ 0 in the vector bundle Q → M induces an isometry (Z^{ϵ}, g_{t}^{ϵ}) → (Z^{s}, g_{t/|s|}^{s}), which preserves the almost ϵ-complex structure, where ϵ = sgn(s). This shows that it is sufficient to consider only three of the above twistor spaces, namely Z^{+} := Z^{+1}, Z^{−} := Z^{−1} and Z^{0}. We will study the integrability of the almost ϵ-complex structure 𝒥^{ϵ} and the holomorphicity of the horizontal distribution H ⊂ T Z^{ϵ}, which is 𝒥^{ϵ}-invariant. For this we extend the G-structure approach developed in [AG] to the para-case (ϵ = 1).

4.3 Twistor spaces of para-quaternionic (Kähler) manifolds as bundles associated to G-structures

In this subsection we interpret the twistor spaces Z^{ϵ} (ϵ = −1, 0, 1) from the point of view of G-structures.

Let (M, Q) be a para-quaternionic manifold. Note that Q̃_{m} := ℝId + Q_{m} ⊂ End T_{m}M is an algebra isomorphic to the algebra of para-quaternions, i.e. to the matrix algebra ℝ(2). Since any irreducible module of ℝ(2) is isomorphic to ℝ^{2}, the Q̃_{m}-module T_{m}M is isomorphic to the ℝ(2)-module ℝ^{2} ⊗ ℝ^{n}, 2n = dim M, with the action on the first factor.

Definition 6 Let (M, Q) be an (almost) para-quaternionic manifold. A para-quaternionic coframe at m ∈ M is an isomorphism ϕ : T_{m}M →^{∼} ℝ^{2} ⊗ ℝ^{n} which maps Q̃_{m} into ℝ(2), i.e.

ϕ ∘ Q̃_{m} ∘ ϕ^{−1} = ℝ(2) ⊗ Id .

Proposition 8

(i) The set P of all para-quaternionic coframes together with the natural projection π^{P} : P → M is a G-structure, i.e. a principal subbundle of the bundle of all coframes with the structure group G := SL_{2}^{±}(ℝ) ⊗ GL_{n}(ℝ), where

SL_{2}^{±}(ℝ) = {A ∈ GL_{2}(ℝ)| det A = ±1} .

(ii) Let A ∈ 𝔰𝔩_{2}(ℝ) ⊗ Id ⊂ 𝔤 = Lie G such that A^{2} = ϵId and G_{A} the stabilizer (i.e. centralizer) of A in G. There is a canonical isomorphism of fibre bundles

P/G_{A} →^{∼} Z^{ϵ} .

Proof: (i) It is clear that any two para-quaternionic coframes are related by an element of GL_{2}(ℝ) ⊗ GL_{n}(ℝ) = SL_{2}^{±}(ℝ) ⊗ GL_{n}(ℝ). (ii) Let ϕ ∈ P be a coframe at m ∈ M. It induces an algebra isomorphism ϕ̂: ℝ(2) → Q̃_{m}, B ↦ ϕ^{−1}Bϕ. The image ϕ̂(A) ∈ Q_{m} satisfies ϕ̂(A)^{2} = ϵId, hence ϕ̂(A) ∈ Z_{m}^{ϵ}. If k ∈ G_{A} then k̂ϕ(A) = ϕ^{−1}k^{−1}Akϕ = ϕ̂(A). So the map P → Z^{ϵ}, ϕ ↦ ϕ̂(A), factorizes to an isomorphism P/G_{A} → Z^{ϵ} of fibre bundles.

Assume now that (M, g, Q) is a para-quaternionic Kähler manifold of dimension 4n, or more generally an almost para-quaternionic Hermitian manifold. On ℝ^{2} ⊗ ℝ^{2n} we fix the standard scalar product g_{can} = ω_{ℝ^{2}} ⊗ ω_{ℝ^{2n}}, where ω_{ℝ^{2n}} denotes the standard symplectic structure of ℝ^{2n}.

Definition 7 Let (M, g, Q) be an almost para-quaternionic Hermitian manifold of dimension 4n. A para-quaternionic Hermitian coframe at m ∈ M is a linear isometry

ϕ : (T_{m}M, g_{m}) →^{∼} (ℝ^{2} ⊗ ℝ^{2n}, g_{can}) which maps Q̃_{m} into ℝ(2).

Proposition 9 The set P of all para-quaternionic Hermitian coframes together with the natural projection π^{P} : P → M is a G-structure with G = G_{0} ∪ ξG_{0}, G_{0} := SL_{2}(ℝ) ⊗ Sp(ℝ^{2n}), ξ = A ⊗ B ∈ SL_{2}^{±}(ℝ) ⊗ GL_{n}(ℝ), det A = −1 and B^{∗}ω_{ℝ^{2n}} = −ω_{ℝ^{2n}}. Moreover, the twistor space Z^{ϵ} is canonically isomorphic to the bundle P/G_{A}, where 0 ≠ A ∈ 𝔰𝔩_{2}(ℝ) with A^{2} = ϵId.

5 G-structures of para-twistor type and their twistor spaces: obstructions for integrability

5.1 Groups of para-twistor type and para-complex symmetric spaces

Definition 8 A connected linear Lie group G ⊂ GL(V), V = ℝ^{n}, is called of paratwistor type if its Lie algebra contains a para-complex structure, i.e. an element J such that J^{2} = Id. (If G is not connected, we shall assume, in addition, that the conjugation by J preserves G.)

Since the endomorphism J is semi-simple, the adjoint operator ad_{J} is semi-simple and, hence, we have the direct sum 𝔤 = 𝔨 + 𝔪, where 𝔨 = ker ad_{J} = Z_{𝔤}(J) and 𝔪 = [J, 𝔤]. It follows that

𝔪 = {A ∈ 𝔤|{J, A} = AJ + JA = 0} .

This implies that [𝔪, 𝔪] ⊂ 𝔨 and, hence, that 𝔤 = 𝔨 + 𝔪 is a symmetric decomposition.

Proposition 10 The orbit S := Ad_{G}(J) ≅ G/K, K := Z_{G}(J), is an affine symmetric space and carries a canonical G-invariant para-complex structure J^{S}.

Proof: The involutive automorphism A ↦ JAJ^{−1} = JAJ of G has K as its fixed point set and defines the symmetry of G/K at the point eK.

The formula J_{𝔪}A = JA = \frac{1}{2}[J, A], A ∈ 𝔪, defines a K-invariant para-complex structure on 𝔪, which extends to a G-invariant para-complex structure J^{S} on S. The structure J^{S} is integrable, since it is parallel under the canonical torsion-free connection of the symmetric space S.

The projections onto 𝔨 and 𝔪 are given by

A ↦ \frac{1}{2}J\{J, A\} = \frac{1}{2}(A + JAJ)  (5.1)
A ↦ \frac{1}{2}J[J, A] = \frac{1}{2}(A − JAJ) .  (5.2)

5.2 The space of curvature tensors

Let G ⊂ GL(V) be a linear Lie group of para-twistor type with Lie algebra 𝔤, J ∈ 𝔤 a paracomplex structure and 𝔤 = 𝔨 + 𝔪 the corresponding symmetric decomposition; 𝔨 = Z_{𝔤}(J) and 𝔪 = [J, 𝔤]. Recall that 𝔪 carries the para-complex structure J_{𝔪} : A ↦ JA = \frac{1}{2}[J, A]. For any subspace U ⊂ End V we denote by

ℛ(U) := {R ∈ U ⊗ ∧^{2}V^{∗}|R satisfies the first Bianchi identity}

the vector space of algebraic curvature tensor of type U.

The projection π_{𝔪} : 𝔤 → 𝔪 induces a projection

π_{𝔪} : C^{2}(𝔤) → C^{2}(𝔪) .

According to (5.2), the projection α^{𝔪} := π_{𝔪}α ∈ C^{2}(𝔪) of α ∈ C^{2}(𝔤) is given by

α^{𝔪}(X, Y) = \frac{1}{2}(α(X, Y) − Jα(X, Y)J) .  (5.3)

Recall that, since 𝔪 ⊂ End V is endowed with the para-complex structure J_{𝔪}, we have the decomposition (2.2)

C^{2}(𝔪) = ∑_{p+q=2} C^{p,q}(𝔪) .

We put π_{𝔪}^{p,q} := π^{p,q} ∘ π_{𝔪} : C^{2}(𝔤) → C^{p,q}(𝔪) and ℛ^{p,q}(𝔪) := ℛ(𝔪) ∩ C^{p,q}(𝔪).

The action of J as an automorphism of the tensor algebra induces involutions

T_{J} : C^{2}(𝔤) → C^{2}(𝔤) , T_{J} : C^{2}(V) → C^{2}(V) .

We denote the ±1-eigenspaces of T_{J} on C^{2}(𝔤) by C_{±}^{2}(𝔤), such that

C^{2}(𝔤) = C_{+}^{2}(𝔤) + C_{−}^{2}(𝔤) ,

and put C_{±}^{2}(U) := C_{±}^{2}(𝔤) ∩ C^{2}(U) and ℛ_{±}(U) := C_{±}^{2}(𝔤) ∩ ℛ(U), where U = 𝔨, 𝔪.

Proposition 11

(i) The eigenspaces of T_{J} on C^{2}(𝔤) are given by

C_{+}^{2}(𝔪) = C^{1,1}(𝔪)  (5.4)
C_{−}^{2}(𝔪) = C^{2,0}(𝔪) + C^{0,2}(𝔪) .  (5.5)

(ii) The action of T_{J} on C^{p,q}(V) is given by

T_{J}α^{1,1} = −Jα^{1,1}
T_{J}α^{2,0} = Jα^{2,0}
T_{J}α^{0,2} = Jα^{0,2} .

In particular,

C^{1,1}(V) = ker(T_{J} + L_{J})
C^{2,0}(V) + C^{0,2}(V) = ker(T_{J} − L_{J}) ,

where L_{J}α = J ∘ α. The action of J as a derivation on the tensor algebra induces an endomorphism

α ↦ J · α = [J, α] − α(J·, ·) − α(·, J·)

of C^{2}(𝔤). Similarly, J acts as a derivation on C^{2}(V). Proposition 12

(i) The action of J as a derivation on C^{2}(𝔤) is given by

J · α^{p,q} = 2qJα^{p,q} for all α^{p,q} ∈ C^{p,q}(𝔪) ,
J · α = −2α(J·, ·) for all α ∈ C_{+}^{2}(𝔨) ,
J · C_{−}^{2}(𝔨) = 0 .

In particular, the vector space of J-invariants is given by

C^{2}(𝔤)^{J} = C_{−}^{2}(𝔨) + C^{2,0}(𝔪) .  (5.6)

(ii) The action of J as a derivation on C^{2}(V) is given by

J · α^{2,0} = −Jα^{2,0}
J · α^{0,2} = 3Jα^{0,2}
J · α^{1,1} = Jα^{1,1} .

In particular,

C^{2,0}(V) = ker(D_{J} + L_{J})
C^{0,2}(V) = ker(D_{J} − 3L_{J})
C^{1,1}(V) = ker(D_{J} − L_{J}) ,

where D_{J}α = J · α.

The proposition shows that π_{𝔪}^{1,1}C^{2}(𝔤)^{J} = π_{𝔪}^{0,2}C^{2}(𝔤)^{J} = 0 and π_{𝔪}^{2,0}C^{2}(𝔤)^{J} = C^{2,0}(𝔪).

Proposition 13 The following holds

(i) ℛ(𝔤) = ℛ_{+}(𝔤) + ℛ_{−}(𝔤),

(ii) ℛ(𝔪) = ℛ_{+}(𝔪) + ℛ_{−}(𝔪),

(iii) π_{𝔪}ℛ_{+}(𝔤) = π_{𝔪}^{1,1}ℛ_{+}(𝔤) ⊃ ℛ_{+}(𝔪) = ℛ^{1,1}(𝔪),

(iv) π_{𝔪}^{0,2}ℛ(𝔤) = ℛ^{0,2}(𝔪),

(v) π_{𝔪}ℛ_{−}(𝔤) = (π_{𝔪}^{2,0} + π_{𝔪}^{0,2})ℛ_{−}(𝔤) ⊃ ℛ_{−}(𝔪) = ℛ^{2,0}(𝔪) + ℛ^{0,2}(𝔪).

Proof: (i) and (ii) follow from the fact that T_{J} : C^{2}(𝔤) → C^{2}(𝔤) preserves the subspaces ℛ(𝔪) ⊂ ℛ(𝔤) ⊂ C^{2}(𝔤) and (iii) follows from the equation (5.4). The equation (5.5) and (iv) imply (v). Therefore it suffices to prove (iv). For R ∈ ℛ(𝔤) and X, Y, Z ∈ V we calculate

(π_{𝔪}^{0,2}R)(X, Y) = \frac{1}{4}(R^{𝔪}(X, Y) + R^{𝔪}(JX, JY) − JR^{𝔪}(JX, Y) − JR^{𝔪}(X, JY))
= \frac{1}{8}(R(X, Y) − JR(X, Y)J + R(JX, JY) − JR(JX, JY)J

−JR(JX, Y) + R(JX, Y)J − JR(X, JY) + R(X, JY)J)

= \frac{1}{8}(R(X, Y) − JR(X, Y)J − JR(JX, Y) − JR(X, JY))

+\frac{1}{8}(−JR(JX, JY)J + R(JX, JY) + R(X, JY)J + R(JX, Y)J)

= \frac{1}{8}(J(J · R)(X, Y) − (J · R)(JX, JY)J)

and, therefore,

∑_{cyclic}(π_{𝔪}^{0,2}R)(X, Y)Z = \frac{1}{8}J ∑_{cyclic}(J · R)(X, Y)Z − \frac{1}{8} ∑_{cyclic}(J · R)(JX, JY)JZ = 0 ,

where the sum is over cyclic permutations of (X, Y, Z). Here we used the fact that A · ℛ(𝔤) ⊂ ℛ(𝔤) for any A ∈ 𝔤𝔩(V).

5.3 G-structures with connection and associated K-structures

Let G ⊂ GL(V), V = ℝ^{n}, be a linear Lie group.

Definition 9 A G-structure on a manifold M is G-principal bundle π : P → M endowed with a displacement form θ, i.e. a G-equivariant V -valued one-form such that ker θ = T^{v}P := ker dπ.

We shall identify a point p ∈ P with the coframe

p : T_{π(p)}M → V , X ↦ θ_{p}((dπ)_{p}^{−1}(X)) .

Definition 10 A principal connection in a G-principal bundle π : P → M is a Gequivariant 𝔤-valued one-form ω : T P → 𝔤 such that H := ker ω is a distribution transversal to the vertical distribution T^{v}P.

Recall that the wedge product of two one-forms α, β with values in a Lie algebra is the Lie algebra valued two-form given by

[α ∧ β](X, Y) := [α(X), β(Y)] − [α(Y), β(X)] .

The curvature of a connection ω is the 𝔤-valued G-equivariant horizontal two-form

Ω = dω + \frac{1}{2}[ω ∧ ω] .

If π : P → M is a G-structure with displacement form θ, then the torsion of ω is the V -valued G-equivariant horizontal two-form

Θ := dθ + [ω ∧ θ] ,

where the Lie bracket is taken in the affine Lie algebra V + 𝔤.

If θ is the displacement form of a G-structure π : P → M and ω a principal connection then:

κ = θ + ω : T P → V ⊕ 𝔤

is a Cartan connection, i.e. a G-equivariant absolute parallelism which extends the canonical vertical parallelism T^{v}P → 𝔤. The curvature of the Cartan connection κ is defined as the (V ⊕ 𝔤)-valued G-equivariant horizontal two-form

Ω_{κ} := dκ + \frac{1}{2}[κ ∧ κ] .

Notice that the V and 𝔤-components of Ω_{κ} are exactly the torsion and curvature forms of ω:

Ω_{κ}^{V} = Θ , Ω_{κ}^{𝔤} = Ω .

Let now K ⊂ G be a Lie subgroup with Lie algebra 𝔨 and 𝔤 = 𝔨 + 𝔪 a K-invariant direct decomposition of the vector space 𝔤. Accordingly, any 𝔤-valued form α on P is decomposed as

α = α^{𝔨} + α^{𝔪} .

Proposition 14 [AG] Let (π : P → M, θ, ω) be a G-structure with a connection and K ⊂ G a Lie subgroup. Then

π′ : P → Z := P/K

is a K-structure with displacement form

θ′ := θ + ω^{𝔪} : T P → V′ := V ⊕ 𝔪

and connection

ω′ := ω^{𝔨} .

The curvature Ω′ and torsion Θ′ of ω′ are given by

Θ′ = (Θ′)^{V} + (Θ′)^{𝔪} = (Θ − [ω^{𝔪} ∧ θ]) + Ω^{𝔪} − \frac{1}{2}[ω^{𝔪} ∧ ω^{𝔪}]^{𝔪}
Ω′ = Ω^{𝔨} − \frac{1}{2}[ω^{𝔪} ∧ ω^{𝔪}]^{𝔨} .

5.4 The twistor space of a G-structure of para-twistor type

Let G ⊂ GL(V) be a linear Lie group of para-twistor type, J ∈ 𝔤 a para-complex structure and 𝔤 = 𝔨+𝔪 the corresponding symmetric decomposition; 𝔨 = Z_{𝔤}(J) and 𝔪 = [J, 𝔤]. Let π : P → M be a G-structure endowed with a principal connection ω : T P → 𝔤. (P, ω) will be called a G-structure of para-twistor type. The vector space V′ := V ⊕ 𝔪 has the para-complex structure J′ = J ⊕ J_{𝔪}. The natural action of K = Z_{G}(J) on V′ preserves this structure and is identified with a subgroup K ⊂ GL(V′, C) := Aut(V′, J′). This implies that the K-structure

π′ : P → Z := P/K

is subordinated to a GL(V′, C)-structure, i.e. to an almost para-complex structure 𝒥 on Z. At the point z = π′p ∈ Z, p ∈ P, the almost para-complex structure 𝒥 is defined by:

𝒥_{z} = p̂^{−1} ∘ J′ ∘ p̂,

where p̂: T_{z}Z → V′ is the coframe associated with p ∈ P. It is easily checked that this definition does not depend on p ∈ (π′)^{−1}(z).

Similarly, we can associate a para-complex structure J_{z} : T_{πp}M → T_{πp}M with any point z = Kp ∈ Z by the formula

J_{z} := p ∘ J ∘ p^{−1} ,

using the isomorphism p : T_{πp}M → V. This allows to identify the G/K-bundle π_{Z} : Z = P/K → M = P/G with a bundle of para-complex structures on the tangent spaces of M.

We denote by ℋ_{Z} = π′_{∗}kerω ⊂ T Z the projection of the horizontal distribution of ω to T Z. We call it the horizontal distribution of Z.

Definition 11 Let (π : P → M, ω) be a G-structure of para-twistor type and K = Z_{G}(J). Then the induced K-structure π′ : P → Z = P/K endowed with the induced connection ω′ = π_{𝔨} ∘ ω, the horizontal distribution ℋ_{Z} and the almost para-complex structure 𝒥 is called the twistor space associated to the G-structure of para-twistor type (P, ω) and to the para-complex structure J ∈ 𝔤.

Notice that the almost para-complex structure 𝒥 and the horizontal distribution ℋ_{Z} are invariant under the parallel transport in T Z defined by the connection ω′. Therefore, we can apply Propositions 1 and 2.

Theorem 1 Let (π : P → M, ω) be a G-structure of para-twistor type, where ω is a principal connection with curvature form Ω and torsion form Θ and (Z, 𝒥, ℋ_{Z}) the corresponding twistor space. Then

(i) The almost para-complex structure 𝒥 on Z is integrable if and only if

π^{0,2} ∘ Θ = 0 and π_{𝔪}^{0,2} ∘ Ω = 0 ,  (5.7)

(ii) The horizontal distribution ℋ_{Z} ⊂ T Z is para-holomorphic if and only if

π_{𝔪}^{1,1} ∘ Ω = 0 ,

where we consider the values of the horizontal forms Θ and Ω at p ∈ P as

Θ_{p} : ∧^{2}T_{π′p}Z → V and Ω_{p} : ∧^{2}T_{π′p}Z → 𝔤 .

Proof: Since G is of para-twistor type, 𝔤 = 𝔨 + 𝔪 is a symmetric decomposition and, in particular, [𝔪, 𝔪] ⊂ 𝔨. By Proposition 14, the torsion of the connection ω′ in the K-principal bundle π′ : P → Z is given by

Θ′ = (Θ′)^{V} + (Θ′)^{𝔪} = (Θ − [ω^{𝔪} ∧ θ]) + Ω^{𝔪} .

The second term [ω^{𝔪} ∧ θ]_{p} : ∧^{2}T_{π′p}Z → V′ = V ⊕ 𝔪, p ∈ P, on the right-hand side is of type (2, 0) since

θ′ = θ + ω^{𝔪} : T_{π′p}Z → V′

is of type (1, 0):

θ′ ∘ 𝒥_{π′p} = (J ⊕ J_{𝔪}) ∘ θ′ .

Therefore the integrability condition π^{0,2}Θ′ = 0 of Proposition 1 reduces to (5.7).

To prove (ii), we notice that the coframe p̂: T_{π′p}Z → V′ = V ⊕ 𝔪 maps the horizontal space (ℋ_{Z})_{π′p} to V. Therefore the tensor

S = T mod ℋ_{Z}

corresponds to (Θ′)^{𝔪} = Ω^{𝔪} and S^{1,1} corresponds to π_{𝔪}^{1,1} ∘ Ω. The two-form Ω^{𝔪} on P vanishes on the vertical distribution T^{v}P = κ^{−1}(𝔤). This implies that π_{𝔪}^{1,1} ∘ Ω vanishes on p̂^{−1}(𝔪). Therefore the para-holomorphicity condition (2.5) of Proposition 2 reduces to π_{𝔪}^{1,1} ∘ Ω|_{ℋ_{Z}×ℋ_{Z}} = 0, which is equivalent to π_{𝔪}^{1,1} ∘ Ω = 0.

Since any p ∈ P is an isomorphism p : T_{πp}M → V we can identify the horizontal two-forms Θ and Ω with G-equivariant functions

T : P → ∧^{2}V^{∗} ⊗ V and R : P → ∧^{2}V^{∗} ⊗ 𝔤 .

In particular, T +π_{𝔪}∘R : P → ∧^{2}V^{∗}⊗V′ = C^{2}(V′) = ⊕C^{p,q}(V′). Now we can reformulate the theorem in terms of T and R_{𝔪} := π_{𝔪} ∘ R.

Corollary 1 Under the assumptions of the previous theorem, the following is true.

(i) The almost para-complex structure is integrable if and only if T and R_{𝔪} take values in C^{2,0}(V′) ⊕ C^{1,1}(V′).

(ii) The horizontal distribution is para-holomorphic if and only if R_{𝔪} takes values in C_{−}(𝔪) = C^{2,0}(𝔪) ⊕ C^{0,2}(𝔪).

Both conditions are satisfied if and only if R_{𝔪} is of type (2, 0) and T is of type (2, 0)+(1, 1).

Now we choose a local section p_{0} : M → P and identify P locally with M × G. We denote by T^{(p_{0})} and R^{(p_{0})} the restrictions of T and R to M = M × {e} ⊂ M × G. Then

T_{(x,g)} = g_{∗}T_{x}^{(p_{0})} = gT_{x}^{(p_{0})}(g^{−1}·, g^{−1}·)

and

R_{(x,g)} = g_{∗}R_{x}^{(p_{0})} = gR_{x}^{(p_{0})}(g^{−1}·, g^{−1}·)g^{−1} .

This implies, for all u, v ∈ V,

π_{𝔪}R_{(x,g)}(u, v) = π_{𝔪}gR_{x}^{(p_{0})}(g^{−1}u, g^{−1}v)g^{−1}
= gπ_{g^{−1}𝔪g}R_{x}^{(p_{0})}(g^{−1}u, g^{−1}v)g^{−1} = g_{∗}(π_{g^{−1}𝔪g}R_{x}^{(p_{0})})(u, v) .

For any para-complex structure I = gJg^{−1} ∈ S = G/K we have the vector spaces 𝔪(I) = [I, 𝔤] = g𝔪g^{−1} and V′(I) = V ⊕ 𝔪(I) with the para-complex structures gJ_{𝔪}g^{−1} and I′ = gJ′g^{−1}, respectively.

The above calculation implies that the (p, q) component of T or R_{𝔪}, with respect to (J, J′), vanishes if and only if the (p, q) component of T^{(p_{0})} or π_{𝔪(I)} ∘ R^{(p_{0})}, with respect to (I, I′), vanishes for all I ∈ S. We we will use the symbol π_{𝔪(I)}^{p,q} := π_{I}^{p,q} ∘ π_{𝔪(I)}, where π_{I}^{p,q} : C^{2}(𝔪(I)) → C_{I}^{p,q}(𝔪(I)) is the projection onto the (p, q)-component with respect to (I, I′) for any I ∈ S. Similarly we define π_{I}^{p,q} : C^{2}(V) → C_{I}^{p,q}(V) as the projection onto the (p, q)-component with respect to I.

This motivates the definition of the following two G-submodules of ℛ(𝔤):

ℛ_{int}(𝔤) := {R ∈ ℛ(𝔤)|π_{𝔪(I)}^{0,2}R = 0 for all I ∈ S}
ℛ_{hol}(𝔤) := {R ∈ ℛ(𝔤)|π_{𝔪(I)}^{1,1}R = 0 for all I ∈ S} .

We also define a G-submodule 𝒯_{int}(𝔤) ⊂ C^{2}(V) by

𝒯_{int}(𝔤) := {T ∈ C^{2}(V)|π_{I}^{0,2}T = 0 for all I ∈ S} .

Corollary 2 Under the assumptions of Theorem 1, the following is true.

(i) The almost para-complex structure 𝒥 is integrable if and only if the functions T^{(p_{0})} and R^{(p_{0})}, associated to a local frame p_{0}, take values in the G-modules 𝒯_{int}(𝔤) and ℛ_{int}(𝔤), respectively.

(ii) The horizontal distribution is para-holomorphic if and only if R^{(p_{0})} takes values in ℛ_{hol}(𝔤).

Both conditions are satisfied if and only if π_{𝔪(I)}R is of type (2, 0) and T is of type (2, 0) + (1, 1) for all I ∈ S.

Corollary 3 Under the assumptions of Theorem 1, the almost para-complex structure 𝒥 on the twistor space Z is integrable and the horizontal distribution ℋ_{Z} is para-holomorhic if for all x ∈ M there exists a frame p ∈ π^{−1}(x) such that the curvature R^{(p)} ∈ ℛ(𝔤) takes values in the G-module

ℛ(𝔤)^{D_{S}} = {R ∈ ℛ(𝔤)|I · R = 0 for all I ∈ S}

and the torsion T^{(p)} satisfies π^{0,2}T^{(p)} = 0.

Proof: This follows from (5.6) and the previous corollary.

Corollary 4 Let G be a group of para-twistor type such that π_{𝔪(I)}ℛ(𝔤) ⊂ C^{2,0}(𝔪(I)), for all I ∈ S, for example if ℛ(𝔤) = ℛ(𝔤)^{D_{S}}. Then for any G-structure (π : P → M, ω) with a torsion-free connection ω, the almost para-complex structure 𝒥 on the twistor space Z is integrable and the horizontal distribution ℋ_{Z} is para-holomorhic.

6 Integrability and holomorphicity results for the twistor spaces of a para-quaternionic Kähler manifold

Theorem 2 Let (M, g, Q) be a para-quaternionic Kähler manifold and (Z^{ϵ}, 𝒥^{ϵ}, ℋ_{Z^{ϵ}}) its twistor space, where ϵ = ±, see sections 4 and 5.4. Then for ϵ = −1 the almost complex structure 𝒥^{ϵ} is integrable and the horizontal distribution is holomorphic. Similarly, for ϵ = 1 the almost para-complex structure 𝒥^{ϵ} is integrable and the horizontal distribution is para-holomorphic.

Proof: By Proposition 9, the para-quaternionic Kähler structure defines a G-structure π : P → M, where G ⊂ GL(ℝ^{2} ⊗ ℝ^{2n}) is the normalizer of the connected Lie group G_{0} := SL_{2}(ℝ) ⊗ Sp(ℝ^{2n}) in SO(2n, 2n). Any para-quaternionic coframe p ∈ P defines an isometry p : (T_{πp}M, g_{πp}) →^{∼} (ℝ^{2} ⊗ ℝ^{2n}, g_{can}), which maps Q_{πp} to 𝔰𝔩(2, ℝ) = 𝔰𝔩(2, ℝ) ⊗ Id, see Definition 7. The linear group G is of para-twistor type and also of twistor type, i.e. there exists elements I, J ∈ 𝔤 = 𝔰𝔩(2, ℝ) ⊕ 𝔰𝔭(ℝ^{2n}) such that I^{2} = −Id and J^{2} = Id. In fact, we can choose I = p ∘ J_{1} ∘ p^{−1} and J = p ∘ J_{2} ∘ p^{−1}. The symmetric space

𝔰𝔩(2, ℝ) = 𝔰𝔩(2, ℝ) ⊗ Id ⊃ S^{−} = Ad_{G}(I) = G/Z_{G}(I) = GL_{2}(ℝ)/Z_{GL_{2}(ℝ)}(I)
= GL_{2}(ℝ)/GL_{1}(ℂ) = SL_{2}^{±}(ℝ)/SO(2)

is the two-sheeted hyperboloid in the three-dimensional Minkowski space 𝔰𝔩(2, ℝ) ≅ ℝ^{2,1}, whereas the symmetric space

𝔰𝔩(2, ℝ) ⊃ S^{+} = Ad_{G}(J) = G/Z_{G}(J) = GL_{2}(ℝ)/Z_{GL_{2}(ℝ)}(J)
= GL_{2}(ℝ)/GL_{1}(C) = SL_{2}(ℝ)/SO(1, 1)

is the one-sheeted hyperboloid.

To finish the proof, in the case ϵ = +1 we apply Corollary 4, in the case ϵ = −1 [AG] Theorem 7.3, since, by Proposition 4, the space of curvature tensors

ℛ(𝔤) = ℛ(𝔤)^{𝔰𝔩(2,ℝ)} = ℛ(𝔤)^{D_{S}} ,

for S = S^{±}.

7 The canonical ϵ-Kähler-Einstein metric and contact structure on the twistor space Z^{ϵ} of a paraquaternionic Kähler manifold

Definition 12 An ϵ-Kähler manifold is a pseudo-Riemannian manifold (M, g) together with a parallel skew-symmetric ϵ-complex structure J. An ϵ-Kähler manifold (M, g, J) is called a Kähler manifold if ϵ = −1 and a para-Kähler manifold if ϵ = +1. The parallel symplectic form ω = g(J·, ·) is called the Kähler form.

Remarks: The metric of a para-Kähler manifold has signature (n, n), since the ±1eigendistributions T^{±}M of J are isotropic. Moreover, they are parallel and ω-Lagrangian.

Conversely, a bi-Lagrangian manifold [Br], i.e. a symplectic manifold (M, ω) with two complementary Lagrangian integrable distributions T^{±}M, has the structure of a para-Kähler manifold, where J|_{T^{±}M} = ±Id and g = ω(J·, ·).

An integrable skew-symmetric ϵ-complex structure on a pseudo-Riemannian manifold is parallel, and hence defines an ϵ-Kähler structure, if and only if the Kähler form ω is closed, see [CMMS1] Thm. 1.

Definition 13 An ϵ-holomorphic distribution 𝒟 of real codimension 2 on an ϵ-complex manifold Z is called an ϵ-holomorphic contact structure if the Frobenius form [·, ·] : ∧^{2}𝒟 → T M/𝒟 is non-degenerate.

Theorem 3 Let (Z^{ϵ}, 𝒥^{ϵ}) be the ϵ-twistor space of a para-quaternionic Kähler manifold (M, g, Q) with non-zero reduced scalar curvature ν. Then

(i) the canonical metric g_{t} = g_{t}^{ϵ} on Z^{ϵ} is ϵ-Kähler-Einstein if and only if t = −\frac{ϵ}{ν}. Moreover, g_{t} is Einstein if and only if t = −\frac{ϵ}{ν} or t = −\frac{ϵ}{ν(n+1)}.

(ii) The horizontal distribution ℋ_{Z} ⊂ T Z^{ϵ} is an ϵ-holomorphic contact structure.

Proof: (i) By Theorem 2 and Proposition 7 the ϵ-complex structure 𝒥^{ϵ} is integrable and g_{t}-skew-symmetric for all t. By the above remark, to check when (Z^{ϵ}, 𝒥^{ϵ}, g_{t}) is ϵ-Kähler it is sufficient to check when the Kähler form ω_{t} = g_{t}(𝒥^{ϵ}·, ·) is closed.

The twistor bundle Z^{ϵ} = P/G_{A} → M, see Proposition 9, is a bundle associated with the principal bundle

P′ := P/Z_{G}(GL_{2}) → M = P′/SO_{3}^{ϵ} ,

where SO_{3}^{ϵ} = SO(2, 1) for ϵ = +1 and SO_{3}^{ϵ} = SO(1, 2) ≅ SO(2, 1) for ϵ = −1. In other words, P′ is the SO_{3}^{ϵ}-principal bundle of standard bases p = (J_{1}, J_{2}, J_{3}) of Q_{x}^{ϵ}, x ∈ M, where J_{1}^{2} = ϵId, J_{2}^{2} = Id and J_{3}^{2} = −ϵId. We have a natural projection

π_{P′} : P′ → Z^{ϵ} = P′/SO_{2}^{ϵ} , (J_{1}, J_{2}, J_{3}) ↦ J_{1} ,

where SO_{2}^{ϵ} = SO(1, 1) for ϵ = +1 and SO_{2}^{ϵ} = SO(2) for ϵ = −1 is the stabilizer of (1, 0, 0)^{t} ∈ ℝ^{3}.

The closure of ω_{t} is equivalent to the closure of its pull back ω′_{t} = π_{P′}^{∗}ω_{t} to P′. The two-form ω′_{t} can be written as

ω′_{t} = g′_{t}(𝒥_{1}·, ·) , g′_{t} = tg^{v} + π_{P′}^{∗}g .  (7.1)

Here π_{P′}^{∗}g is the pull back of the metric g on M and g^{v} is the metric on the vertical bundle T^{v}P′, which corresponds to a suitably normalized ad-invariant scalar product ⟨·, ·⟩ on 𝔰𝔬_{3}^{ϵ} = Lie SO_{3}^{ϵ}, extended by zero to the horizontal bundle ℋ associated with the Levi-Civita connection of M. The normalization of the scalar product ⟨·, ·⟩ on 𝔰𝔬_{3}^{ϵ} = ad(𝔰𝔩_{2}(ℝ)) ≅ 𝔰𝔩_{2}(ℝ) = span{J_{1}^{0}, J_{2}^{0}, J_{3}^{0}} is given by

−ϵ⟨ad_{J_{α}^{0}}, ad_{J_{β}^{0}}⟩ = −4ϵ_{a}δ_{αβ} = 4⟨J_{α}^{0}, J_{β}^{0}⟩ ,  (7.2)

where (J_{1}^{0}, J_{2}^{0}, J_{3}^{0}) is the standard ϵ-quaternionic basis of 𝔰𝔩_{2}(ℝ), with the relations

(J_{α}^{0})^{2} = ϵ_{α}Id , (ϵ_{1}, ϵ_{2}, ϵ_{3}) = (ϵ, 1, −ϵ) .  (7.3)

The above scalar product on 𝔰𝔬_{3}^{ϵ} has signature (2, 1) if ϵ = +1 and (1, 2) if ϵ = −1. The factor 4 is chosen such that the canonical projection (P′, g′_{t}) → (Z^{ϵ} = P′/SO_{2}^{ϵ}, g_{t}) is a pseudo-Riemannian submersion. Notice that the vertical vectors

ad_{J_{2}} , ad_{J_{3}} ∈ T_{p}^{v}P′ ≅ 𝔰𝔬_{3}^{ϵ} = ad(𝔰𝔩_{2}(ℝ)) , p ∈ P′ ,

are mapped to

ad_{J_{2}}J_{1} = −2J_{3} , ad_{J_{3}}J_{1} = −2ϵJ_{2} ∈ T^{v}Z^{ϵ} ⊂ Q_{x}^{ϵ} ≅ 𝔰𝔩_{2}(ℝ) , x = π_{P′}(p) .

The field p ↦ (𝒥_{α})_{p} is defined at p = (J_{1}, J_{2}, J_{3}) as the following endomorphism of T_{p}P′ = T^{v}P′ ⊕ ℋ_{p} ≅ 𝔰𝔬_{3}^{ϵ} ⊕ T_{x}M, x = π_{P′}(p),

𝒥_{α}|_{ℋ_{p}} : T_{x}M → T_{x}M , X ↦ J_{α}X , 𝒥_{α}|_{T^{v}P′} = \frac{1}{2}ad_{J_{α}^{0}} .

It is sufficient to check dω′_{t} = 0 on three vectors, each of which are horizontal or vertical. Moreover, it is sufficient to consider the fundamental vertical fields (V_{1}, V_{2}, V_{3}), which correspond to (J_{1}^{0}, J_{2}^{0}, J_{3}^{0}) and basic horizontal fields X, Y, Z, ... on P′, i.e. horizontal lifts of vector fields X_{M}, Y_{M}, Z_{M} on M.

Lemma 2 With the above notations we have

(i) [V_{1}, V_{2}] = 2V_{3}, [V_{3}, V_{1}] = −2ϵV_{2}, [V_{2}, V_{3}] = −2V_{1},

(ii) the functions g′_{t}(V_{α}, V_{β}) and ω′_{t}(V_{α}, V_{β}) are constant for all α, β ∈ {1, 2, 3},

(iii) [V_{α}, X] = 0,

(iv) [X, Y]^{v} = −\frac{ν}{2} ∑_{α} ϵ_{α}g′_{t}(𝒥_{α}X, Y)V_{α} = −\frac{ν}{2} ∑_{α} ϵ_{α}g(J_{α}X_{M}, Y_{M})V_{α}, where [X, Y]^{v} is

evaluated at the point p = (J_{1}, J_{2}, J_{3}) ∈ P′,

(v) ℒ_{V_{α}}g′_{t} = 0, ℒ_{V_{1}}𝒥_{1} = 0, ℒ_{V_{2}}𝒥_{1} = −2𝒥_{3}, ℒ_{V_{3}}𝒥_{1} = −2ϵ𝒥_{2} and

(vi) (ℒ_{X}g′_{t})(U, V) = 0 for all U, V ∈ T^{v}P′.

Proof: (i) follows from the ϵ-quaternionic relations

[J_{1}^{0}, J_{2}^{0}] = 2J_{3}^{0} , [J_{3}^{0}, J_{1}^{0}] = −2ϵJ_{3}^{0} , [J_{2}^{0}, J_{3}^{0}] = −2J_{1}^{0} .

(ii) Since the metric g^{v} corresponds to the ad-invariant scalar product (7.2), the functions

g′_{t}(V_{α}, V_{β}) = tg^{v}(V_{α}, V_{β}) = −4ϵt⟨J_{α}^{0}, J_{β}^{0}⟩ = 4ϵtϵ_{α}δ_{αβ}

are constant. Similarly, the functions ω′_{t}(V_{α}, V_{β}) are constant, because, for all fundamental vector fields V_{α}, the vector field 𝒥_{1}V_{α} is again a fundamental vector field.

(iii) The vector field [V_{α}, X] is horizontal, since the principal action preserves the horizontal distribution. On the other hand, it is mapped to [0, X_{M}] = 0 under the projection P′ → M. This shows that [V_{α}, X] = 0.

(iv) follows from Proposition 4, since [X, Y]^{v} = −Ω′(X, Y), where Ω′ stands for the curvature form of the principal bundle P′ → M.

(v) ℒ_{V_{α}}g′_{t} = 0 follows from the ad-invariance of g^{v}, cf. (7.1). The remaining equations are obtained from (i) using

𝒥_{1}V_{1} = 0 , 𝒥_{1}V_{2} = V_{3} , 𝒥_{1}V_{3} = ϵV_{2} .

Finally, (ii) and (iii) easily imply (vi).

Part (i) and (ii) of the lemma, yields

dω′_{t}(V_{1}, V_{2}, V_{3}) = V_{1}ω′_{t}(V_{2}, V_{3}) − ω′_{t}([V_{1}, V_{2}], V_{3}) + cycl. = 0 .

Using part (i), (ii), (v) and (vi) of the lemma, we calculate

dω′_{t}(V_{α}, V_{β}, X) = −ω′_{t}([V_{β}, X], V_{α}) − ω′_{t}([X, V_{α}], V_{β}) = −(ℒ_{X}ω′_{t})(V_{α}, V_{β})
= −g′_{t}((ℒ_{X}𝒥_{1})V_{α}, V_{β}) = −g′_{t}([X, 𝒥_{1}V_{α}] − 𝒥_{1}[X, V_{α}], V_{β})
= g′_{t}(X, [V_{β}, 𝒥_{1}V_{α}]) + g′_{t}(X, [𝒥_{1}V_{β}, V_{α}]) = 0 .

By (iii), (iv) and (v) of the lemma, we compute

dω′_{t}(V_{1}, X, Y) = V_{1}ω′_{t}(X, Y) − ω′_{t}([X, Y], V_{1})
= g′_{t}((ℒ_{V_{1}}𝒥_{1})X, Y) + \frac{ν}{2} ∑_{α=1}^{3} ϵ_{α}g(J_{α}X_{M}, Y_{M})ω′_{t}(V_{α}, V_{1})

= 0 + \frac{ν}{2} ∑_{α=1}^{3} ϵ_{α}g(J_{α}X_{M}, Y_{M})g′_{t}(𝒥_{1}V_{α}, V_{1}) = 0 ,

since 𝒥_{1}T^{v}P′ = span{V_{2}, V_{3}}. Similarly, we calculate

dω′_{t}(V_{2}, X, Y) = V_{2}ω′_{t}(X, Y) − ω′_{t}([X, Y], V_{2})
= g′_{t}((ℒ_{V_{2}}𝒥_{1})X, Y) + \frac{ν}{2} ∑_{α=1}^{3} ϵ_{α}g(J_{α}X_{M}, Y_{M})ω′_{t}(V_{α}, V_{2})

= −2g′_{t}(𝒥_{3}X, Y) + \frac{ν}{2} ∑_{α=1}^{3} ϵ_{α}g(J_{α}X_{M}, Y_{M})g′_{t}(𝒥_{1}V_{α}, V_{2})
= −2g(J_{3}X_{M}, Y_{M}) + \frac{νt}{2}ϵ_{3}g(J_{3}X_{M}, Y_{M})g^{v}(𝒥_{1}V_{3}, V_{2})
= −2g(J_{3}X_{M}, Y_{M}) + \frac{νt}{2}(−ϵ)g(J_{3}X_{M}, Y_{M})g^{v}(ϵV_{2}, V_{2})
= −2g(J_{3}X_{M}, Y_{M}) − 2ϵνtg(J_{3}X_{M}, Y_{M}) ,

since g^{v}(V_{2}, V_{2}) = −4ϵ⟨J_{2}^{0}, J_{2}^{0}⟩ = 4ϵϵ_{2} = 4ϵ. In the same way, we obtain

dω′_{t}(V_{3}, X, Y) = −2ϵg(J_{2}X_{M}, Y_{M}) + \frac{νt}{2}g(J_{2}X_{M}, Y_{M})g^{v}(𝒥_{1}V_{2}, V_{3})
= −2ϵg(J_{2}X_{M}, Y_{M}) + \frac{νt}{2}g(J_{2}X_{M}, Y_{M})g^{v}(V_{3}, V_{3})
= −2ϵg(J_{2}X_{M}, Y_{M}) − 2νtg(J_{2}X_{M}, Y_{M}) ,

since g^{v}(V_{3}, V_{3}) = −4ϵ⟨J_{3}^{0}, J_{3}^{0}⟩ = 4ϵϵ_{3} = −4. This shows that dω′_{t}(U, X, Y) = 0 for all vertical vector fields U if and only if νt = −ϵ.

It remains to check that dω′_{t}(X_{p}, Y_{p}, Z_{p}) vanishes on three horizontal vectors

X_{p}, Y_{p}, Z_{p} ∈ ℋ_{p} , p ∈ P′ .

Let t ↦ c̃(t) = (J_{1}(t), J_{2}(t), J_{3}(t)) ∈ P′ be the horizontal lift of a curve t ↦ c(t) ∈ M such that c̃(0) = p and c̃′(0) = X_{p}. Notice that the horizontality of c̃ means that t ↦ J_{α}(t) is parallel along c.

Let t ↦ Y (t) ∈ ℋ_{c̃(t)} be the horizontal lift of the vector field

t ↦ Y_{M}(t) := ∥_{c(0)}^{c(t)}dπ_{P′}Y_{p} ∈ T_{c(t)}M ,

which is parallel along the base curve c. The initial value of Y is Y (0) = Y_{p}. It suffices to prove that

(∇′_{X_{p}}ω′_{t})(Y_{p}, Z_{p}) = g′_{t}((∇′_{X_{p}}𝒥_{1})Y_{p}, Z_{p}) = 0 ,

where ∇′ is the Levi-Civita connection of g′_{t}. We have to check that the horizontal component of

(∇′_{X_{p}}𝒥_{1})Y_{p} = ∇′_{X_{p}}(𝒥_{1}Y) − 𝒥_{1}∇′_{X_{p}}Y

vanishes. Therefore, we calculate

dπ_{P′}(∇′_{X_{p}}(𝒥_{1}Y) − 𝒥_{1}∇′_{X_{p}}Y) = ∇_{c′(0)}(J_{1}(t)Y_{M}(t)) − J_{1}(0)∇_{c′(0)}Y_{M}(t)
= (∇_{c′(0)}J_{1}(t))Y_{M}(0) = 0 .

Here we have used two facts: first, that t ↦ 𝒥_{1}Y (t) is a basic horizontal vector field along c̃, which projects onto

dπ_{P′}𝒥_{1}Y (t) = J_{1}(t)Y_{M}(t)

and, second, that dπ_{P′}∇′_{X}Y = ∇_{X_{M}} Y_{M} for any two basic horizontal vector fields X, Y (e.g. along a horizontal curve), where ∇ is the Levi-Civita connection in M. The latter is a standard fact about pseudo-Riemannian submersions. This proves that g_{t} is ϵ-Kähler-Einstein if and only if t = −\frac{ϵ}{ν}. The above argument proves also the following proposition.

Proposition 15 For any horizontal vectors X, Y, Z on P′ and α = 1, 2, 3, we have

g′_{t}(∇_{X}𝒥_{α})Y, Z) = 0 .

Next we study the Einstein equations for the family g′_{t}. We recall the definition of the O’Neill tensor and the O’Neill formulas for the covariant derivative of a pseudo-Riemannian submersion π : E → M with totally geodesic fibres, see [O, Be]. The O’Neill tensor A ∈ Ω^{1}(End T E) is a one-form with values in skew-symmetric endomorphisms. It is given by

A_{U} = 0 , A_{X}Y = −A_{Y} X = (∇_{X}Y)^{v} = \frac{1}{2}[X, Y]^{v} , A_{X}U = (∇_{X}U)^{h} ,  (7.4)

where U is a vertical vector field and X, Y are horizontal vector fields. The superscripts v and h stand for the vertical and horizontal components, respectively. If X is a basic horizontal vector field then, in addition

A_{X}U = (∇_{X}U)^{h} = ∇_{U}X .  (7.5)

The covariant derivatives in E are given by

∇_{U}V = ∇_{U}^{F}V  (7.6)
∇_{U}X = (∇_{U}X)^{h}  (7.7)
∇_{X}U = (∇_{X}U)^{v} + A_{X}U  (7.8)
∇_{X}Y = A_{X}Y + (∇_{X}Y)^{h} .  (7.9)

Here ∇^{F} and ∇^{M} denote the covariant derivative in the fibres F and in the base M, respectively. For basic horizontal vector fields X, Y, we have [U, X]^{h} = 0 for any vertical (and hence projectable) vector field U. Moreover, we have

(∇_{X}U)^{v} = [X, U]  (7.10)
π_{∗}∇_{X}Y = ∇_{π_{∗}X}^{M}π_{∗}Y .  (7.11)

In particular, ∇_{X}Y is a projectable vector field on E.

Proposition 16 (cf. [O]) Let π : E → M be a pseudo-Riemannian submersion with totally geodesic fibres F. Then the Ricci and scalar curvatures of E are given by:

Ric(U, V) = Ric^{F} (U, V) + ∑_{i} ϵ_{i}⟨A_{X_{i}}U, A_{X_{i}}V ⟩  (7.12)
Ric(X, U) = ⟨(divA)X, U⟩ = ∑_{i} ϵ_{i}⟨(∇_{X_{i}}A)_{X_{i}}X, U⟩  (7.13)
Ric(X, Y) = Ric^{M}(π_{∗}X, π_{∗}Y) − 2 ∑_{i} ϵ_{i}⟨A_{X}X_{i}, A_{Y} X_{i}⟩  (7.14)
scal = π^{∗}scal^{M} + scal^{F} − ∑_{i,j} ϵ_{i}ϵ_{j}⟨A_{X_{i}}X_{j}, A_{X_{i}}X_{j}⟩ .  (7.15)

Proposition 17 The divergence divA ∈ Γ(End T P′) of the O’Neill tensor of the principal bundle P′ → M preserves the horizontal distribution. In particular,

Ric(X, U) = g′_{t}((divA)X, U) = 0 .

Proof: By (7.4) and Lemma 2 (iv), the value of the O’Neill tensor on two basic horizontal vector fields X, Y is given by

A_{X}Y = \frac{1}{2}[X, Y]^{v} = −\frac{ν}{4} ∑_{α} ϵ_{α}g′_{t}(𝒥_{α}X, Y)V_{α} .  (7.16)

It is sufficient to prove that g′_{t}((∇_{X}A)_{Y} Z, U) = 0. This follows from the remark that ∇_{X}V_{α} = A_{X}V_{α} is horizontal, by (7.5), and Proposition 15.

The skew-symmetry of A_{X} and (7.16) imply

A_{X}U = \frac{ν}{4} ∑_{α} ϵ_{α}g′_{t}(U, V_{α})𝒥_{α}X .  (7.17)

In fact,

g′_{t}(A_{X}U, Y) = −g′_{t}(U, A_{X}Y) = \frac{ν}{4} ∑_{α} ϵ_{α}g′_{t}(𝒥_{α}X, Y)g′_{t}(U, V_{α}) .

Proposition 18 Let P′ be the total space of the principal bundle P′ → M of admissible frames of Q over a para-quaternionic Kähler manifold (M, g, Q). Then the Ricci curvature of the metric g′_{t} on P′ is given by:

Ric(U, V) = −ϵ(\frac{1}{2t} + ν^{2}nt)g′_{t}(U, V) , U, V ∈ T^{v}P′  (7.18)
Ric(U, X) = 0 ,  (7.19)
Ric(X, Y) = (ν(n + 2) + \frac{3ϵν^{2}t}{2})g′_{t}(X, Y) , X, Y ∈ ℋ = (T^{v}P′)^{⊥}  (7.20)

Proof: We calculate the Ricci curvature using the formulas in Proposition 16. The fibre F is identified with the Lie group SO(2, 1) with a bi-invariant pseudo-Riemannian metric g′_{t}, which is related with the Killing form B by

g′_{t} = \frac{ϵt}{2}B ,  (7.21)

see (7.2). Therefore

Ric^{F} = −\frac{1}{4}B = \frac{−ϵ}{2t}g′_{t} .  (7.22)

We compute the second term in equation (7.12) using (7.17):

∑_{i} ϵ_{i}⟨A_{X_{i}}U, A_{X_{i}}V ⟩ = ∑_{i} ϵ_{i} \frac{ν^{2}}{16} ∑_{α} ϵ_{α}^{2}g′_{t}(U, V_{α})g′_{t}(V, V_{α})g′_{t}(𝒥_{α}X_{i}, 𝒥_{α}X_{i})

= \frac{ν^{2}}{16} ∑_{i,α} ϵ_{i}^{2}(−ϵ_{α})g′_{t}(U, V_{α})g′_{t}(V, V_{α}) = −ϵν^{2}ntg′_{t}(U, V) .

This implies the first equation (7.18). The second equation (7.19) was already established in Proposition 17. Since M is an Einstein manifold with scalar curvature scal^{M} = 4n(n + 2)ν,

Ric^{M} = \frac{scal}{4n}g = ν(n + 2)g .  (7.23)

We compute the second term in equation (7.14) using (7.16):

−2 ∑_{i} ϵ_{i}⟨A_{X}X_{i}, A_{Y} X_{i}⟩ = −2 ∑_{i,α} ϵ_{i} \frac{ν^{2}}{16}g′_{t}(𝒥_{α}X, X_{i})g′_{t}(𝒥_{α}Y, X_{i})g′_{t}(V_{α}, V_{α})

= −\frac{ν^{2}}{8} ∑_{α} g′_{t}(𝒥_{α}X, 𝒥_{α}Y)g′_{t}(V_{α}, V_{α})

= −\frac{ν^{2}}{8} ∑_{α}(−ϵ_{α})g′_{t}(X, Y)(4ϵtϵ_{α})

= \frac{3ϵν^{2}t}{2}g′_{t}(X, Y) .

This proves the proposition.

Corollary 5 Let P′ be the total space of the principal bundle P′ → M of admissible frames of Q over a para-quaternionic Kähler manifold (M, g, Q) with reduced scalar curvature ν. Then the metric g′_{t} is Einstein if and only if

t = \frac{−ϵ}{ν} or t = \frac{−ϵ}{ν(2n + 3)} .

The corresponding Einstein constant is, respectively,

c = (n + \frac{1}{2})ν and c = \frac{4n^{2} + 14n + 9}{4n + 6}ν .

Next we calculate the Ricci curvature of the metric g_{t}^{ϵ} on the twistor spaces Z^{ϵ} = P′/SO_{2}^{ϵ}, ϵ = ±1.

Proposition 19

(A_{X}Y)_{J_{1}} = \frac{ν}{2}(g(J_{2}π_{∗}X, π_{∗}Y)J_{3} − g(J_{3}π_{∗}X, π_{∗}Y)J_{2}) ∈ T_{J_{1}}^{v}Z = span\{J_{2}, J_{3}\}(7.24)
A_{X}J_{2} = −ϵ_{2} \frac{νt}{2}{ \atop J_{3}π_{∗}X} = −\frac{νt}{2}{ \atop J_{3}π_{∗}X}  (7.25)
A_{X}J_{3} = ϵ_{3} \frac{νt}{2}{ \atop J_{2}π_{∗}X} = −ϵ\frac{νt}{2}{ \atop J_{2}π_{∗}X} ,  (7.26)

where X and Y are horizontal vectors and X̃_{M} ∈ T_{J_{1}}Z^{ϵ} denotes the horizontal lift of the vector X_{M} ∈ T_{π(J_{1})}M.

Ric(U, V) = −ϵ(\frac{1}{t} + ν^{2}nt)g_{t}^{ϵ}(U, V)  (7.27)
Ric(X, U) = 0  (7.28)
Ric(X, Y) = (ν(n + 2) + ϵν^{2}t)g_{t}^{ϵ}(X, Y)  (7.29)

where U and V are vertical vectors.

Proof: The equations (7.24–7.26) are obtained from (7.16), (7.17) and (7.21). We calculate the Ricci curvature using the formulas in Proposition 16. In fact, the projection π : Z^{ϵ} → M is a pseudo-Riemannian submersion with totally geodesic fibre F = SO_{3}^{ϵ}/SO_{2}^{ϵ}, where SO_{3}^{ϵ} ≅ SO(2, 1) and SO_{2}^{ϵ=+1} = SO(1, 1) and SO_{2}^{ϵ=−1} = SO(2). Here Z_{x}^{ϵ} ⊂ Q_{x}^{ϵ} = span{J_{1}, J_{2}, J_{3}}, where (J_{1}, J_{2}, J_{3}) is an admissible basis such that J_{α}^{2} = ϵ_{α}Id and (ϵ_{1}, ϵ_{2}, ϵ_{3}) = (ϵ, 1, −ϵ). In both cases, the Lie algebra 𝔰𝔬_{2}^{ϵ} = ℝ ad (J_{1}). The fibre F is a two-dimensional symmetric space, with symmetric decomposition

𝔰𝔬_{3}^{ϵ} = 𝔰𝔬_{2}^{ϵ} + 𝔪 , 𝔪 = ℝ ad (J_{2}) + ℝ ad (J_{3}) .

The curvature tensor is given by

R(ad (J_{2}), ad (J_{3})) = −ad_{[J_{2},J_{3}]}|_{𝔪} = 2ad_{J_{1}}|_{𝔪}

and the sectional curvature of the metric g^{F} = g_{t}^{ϵ}|_{F} = tg^{v} is \frac{−ϵ}{t}. In particular,

Ric^{F} = −ϵg^{v} = −\frac{ϵ}{t}g^{F} .  (7.30)

Next we compute the second term in equation (7.12) using (7.25):

∑_{i} ϵ_{i}g_{t}^{ϵ}(A_{X_{i}}J_{2}, A_{X_{i}}J_{2}) = \frac{ν^{2}t^{2}}{4} ∑_{i} ϵ_{i}g(J_{3}π_{∗}X_{i}, J_{3}π_{∗}X_{i})
= ν^{2}t^{2}n(−ϵ_{3}) = ϵν^{2}t^{2}n
= −ϵν^{2}tng_{t}^{ϵ}(J_{2}, J_{2}) ,

since g_{t}^{ϵ}(J_{2}, J_{2}) = −tϵ_{2} = −t. The same calculation for (U, V) = (J_{2}, J_{3}) and (U, V) = (J_{3}, J_{3}) shows that for any two vertical vectors U, V, we have

∑_{i} ϵ_{i}g_{t}^{ϵ}(A_{X_{i}}U, A_{X_{i}}V) = −ϵν^{2}tng_{t}^{ϵ}(U, V) .

This proves (7.27).

Now we calculate the second term in equation (7.14) using (7.24).

−2 ∑_{i} ϵ_{i}g_{t}^{ϵ}(A_{X}X_{i}, A_{Y} X_{i})

= −\frac{ν^{2}}{2} ∑_{i} ϵ_{i}g(J_{2}π_{∗}X, π_{∗}X_{i})g(J_{2}π_{∗}Y, π_{∗}X_{i})g_{t}^{ϵ}(J_{3}, J_{3})

−\frac{ν^{2}}{2} ∑_{i} ϵ_{i}g(J_{3}π_{∗}X, π_{∗}X_{i})g(J_{3}π_{∗}Y, π_{∗}X_{i})g_{t}^{ϵ}(J_{2}, J_{2})

= −\frac{ν^{2}}{2} g(J_{2}π_{∗}X, J_{2}π_{∗}Y)g_{t}^{ϵ}(J_{3}, J_{3}) − \frac{ν^{2}}{2} g(J_{3}π_{∗}X, J_{3}π_{∗}Y)g_{t}^{ϵ}(J_{2}, J_{2})
= −\frac{ν^{2}}{2} [(−ϵ_{2})(−tϵ_{3}) + (−ϵ_{3})(−tϵ_{2})]g_{t}^{ϵ}(X, Y) = ϵν^{2}tg_{t}^{ϵ}(X, Y) .

This proves (7.29).

To prove that Ric(X, U) = 0, by Proposition 16 we have to check that divA preserves the horizontal distribution ℋ_{Z} ⊂ T Z^{ϵ}. It sufficient to prove that

g_{t}^{ϵ}((∇_{X}A)_{Y} Z, 𝒥^{ϵ}U) = 0

for all basic horizontal vector fields X, Y, Z and vertical vector fields U. We compute this using the fact that ∇𝒥^{ϵ} = 0 and (7.10):

g_{t}^{ϵ}((∇_{X}A)_{Y} Z, 𝒥^{ϵ}U) = Xg_{t}^{ϵ}(A_{Y} Z, 𝒥^{ϵ}U) − g_{t}^{ϵ}(A_{Y} Z, 𝒥^{ϵ}∇_{X}U)
= Xg_{t}^{ϵ}(A_{Y} Z, 𝒥^{ϵ}U) − g_{t}^{ϵ}(A_{Y} Z, 𝒥^{ϵ}[X, U]) .

Lemma 3 For any basic horizontal vector fields X, Y and vertical vector field U we have

g_{t}^{ϵ}(A_{X}Y, 𝒥^{ϵ}U) = \frac{ϵνt}{2}g(Uπ_{∗}X, π_{∗}Y) = −\frac{1}{2}Uω_{t}(X, Y) ,  (7.31)

where ω_{t} = g_{t}^{ϵ}(𝒥^{ϵ}·, ·) is the ϵ-Kähler form and the value U_{J_{1}} ∈ T_{J_{1}}^{v}Z = span{J_{2}, J_{3}} ⊂ Q_{x}, x = π(J_{1}), of the vertical vector field U at the point J_{1} ∈ Z^{ϵ} is considered as an endomorphism of T_{x}M.

Proof: The first equation follows from (7.24) and the formulas 𝒥^{ϵ}J_{2} = J_{3}, 𝒥^{ϵ}J_{3} = ϵJ_{2}. For the second equality we use that [U, X] and [U, Y] are vertical and that ω_{t} is closed:

ℒ_{U}(ω_{t}(X, Y)) = (ℒ_{U}ω_{t})(X, Y) = (dι_{U}ω_{t})(X, Y)
= −ω_{t}(U, [X, Y]) = −2g_{t}^{ϵ}(𝒥^{ϵ}U, A_{X}Y) .

The following corollary finishes the proof of Theorem 3 (i).

Corollary 6 Let Z^{ϵ}, ϵ = ±1, be the twistor spaces of a para-quaternionic Kähler manifold. Then the metric g_{t}^{ϵ} is Einstein if and only if

t = −\frac{ϵ}{ν} or t = −\frac{ϵ}{ν(n + 1)} .

The corresponding Einstein constant is, respectively,

c = (n + 1)ν and c = \frac{n^{2} + 3n + 1}{n + 1}ν .

(ii) By Theorem 2, we know that the horizontal distribution ℋ_{Z} ⊂ T Z^{ϵ} is holomorphic if ϵ = −1 and para-holomorphic if ϵ = −1. We show that it is a para-holomorphic contact structure if ϵ = +1. The case ϵ = −1 is similar. We have to check that the Frobenius form

ℋ_{Z}^{1,0} × ℋ_{Z}^{1,0} ∋ (Z, W) ↦ ([Z, W] mod ℋ_{Z}^{1,0}) ∈ T^{1,0}Z^{ϵ}/ℋ_{Z}^{1,0}

of ℋ_{Z}^{1,0} is nondegenerate.

Let X and Y be basic horizontal vector fields on P′ and Z = X + e𝒥_{1}X and W = Y + e𝒥_{1}Y the corresponding sections of ℋ^{1,0} ⊂ ℋ ⊗ C = ℋ + eℋ the (+e)-eigenbundle of the C-linear extension of 𝒥_{1} on ℋ ⊗ C. Notice that 𝒥_{1}^{2}|_{ℋ} = ϵId = Id, since ϵ = +1. Let us calculate, with the help of part (iv) of Lemma 2, the vertical component of [Z, W] at any point p = (J_{1}, J_{2}, J_{3}) ∈ P′:

[Z, W]^{v} = −\frac{ν}{2} ∑_{α} ϵ_{α}(g(J_{α}X_{M}, Y_{M}) + g(J_{α}J_{1}X_{M}, J_{1}Y_{M}))V_{α}

−e\frac{ν}{2} ∑_{α} ϵ_{α}(g(J_{α}X_{M}, J_{1}Y_{M}) + g(J_{α}J_{1}X_{M}, Y_{M}))V_{α}

= −ν(ρ_{2}(X_{M}, Y_{M})V_{2} − ρ_{3}(X_{M}, Y_{M})V_{3})

+eν(ρ_{3}(X_{M}, Y_{M})V_{2} − ρ_{2}(X_{M}, Y_{M})V_{3})

= −ν(ρ_{2}(X_{M}, Y_{M}) − eρ_{3}(X_{M}, Y_{M}))(V_{2} + eV_{3}) ,

where ρ_{α} = g(J_{α}·, ·). This shows that the Frobenius form of ℋ^{1,0} ⊂ T P′ ⊗ C is nondegenerate. Let us denote by X̃_{M} and Ỹ_{M} the horizontal lifts of X_{M} and Y_{M} to vector fields on Z^{ϵ}. We put Z̃ := X_{M} + e𝒥^{ϵ}X_{M} and W̃ := Y_{M} + e𝒥^{ϵ}Y_{M}. Thanks to the above formula, we can calculate the vertical component of [Z̃, W̃] at the point z = J_{1} ∈ Z^{ϵ}, which is the image of p = (J_{1}, J_{2}, J_{3}) ∈ P′ under the natural projection P′ → Z^{ϵ} = P′/SO_{2}^{ϵ}.

[Z̃, W̃]^{v} = −ν(ρ_{2}(X_{M}, Y_{M}) − eρ_{3}(X_{M}, Y_{M}))([J_{2}, J_{1}] + e[J_{3}, J_{1}])
= 2ν(ρ_{2}(X_{M}, Y_{M}) − eρ_{3}(X_{M}, Y_{M}))(J_{3} + eJ_{2}) .

This shows that ℋ_{Z} ⊂ T Z^{ϵ} is a para-holomorphic contact structure if ϵ = +1.

8 Twistor construction of minimal submanifolds of para-quaternionic Kähler manifolds

8.1 Kähler and para-Kähler submanifolds of para-quaternionic Kähler manifolds

Definition 14 Let (M, g, Q) be a para-quaternionic Kähler manifold of dimension 4n. An ϵ-Kähler submanifold (ϵ = ±1) of M is a triple (N, J^{ϵ}, g_{N}), where N is a 2mdimensional g-nondegenerate submanifold of M, g_{N} = g|_{N} is the induced pseudo-Rieman-nian metric and J^{ϵ} is a parallel section of the para-quaternionic bundle Q|_{N} such that J^{ϵ}T N = T N and (J^{ϵ})^{2} = ϵId. For ϵ = −1 (M, J^{ϵ}, g_{N}) is called also a Kähler submanifold and for ϵ = +1 it is called a para-Kähler submanifold.

We shall include J^{ϵ} into a local frame (J_{1} = J^{ϵ}, J_{2}, J_{3} = J_{1}J_{2} = −J_{2}J_{1}) of Q|_{N} such that J_{2}^{2} = Id. Such frames (J_{α}) will be called adapted to the ϵ-Kähler submanifold N ⊂ M.

Proposition 20 Let Let (M, g, Q) be a para-quaternionic Kähler manifold of dimension 4n with non-zero reduced scalar curvature ν and N a g-nondegenerate submanifold of M endowed with a section J^{ϵ} ∈ Γ(N, Q) such that (J^{ϵ})^{2} = ϵId and J^{ϵ}T N = T N. Let (J_{α}) be a standard local basis of Q such that J_{1}|_{N} = J^{ϵ}. Then the triple (N, J^{ϵ}, g_{N}) is an ϵ-Kähler submanifold if and only if ω_{2}|_{N} = ω_{3}|_{N} = 0 or, equivalently, J_{2}T N ⊥ T N. In particular, the dimension of an ϵ-Kähler submanifold N ⊂ M is at most 2n.

Proof: It is clear that J_{1} is parallel if and only if ω_{2}|_{N} = ω_{3}|_{N} = 0, see (3.2). Moreover, if ω_{2}|_{N} = ω_{3}|_{N} = 0, then, by the structure equation (3.3), we have that ρ_{2}|_{N} = ρ_{3}|_{N} = 0. Conversely, assume that J_{2}T N ⊥ T N, i.e. ρ_{2}|_{N} = ρ_{3}|_{N} = 0. Differentiating the structure equations for ρ_{2} and ρ_{3}, we get

νdρ′_{α} = −ϵ_{α}νρ′_{β} ∧ ω_{γ} + ϵ_{α}νω_{β} ∧ ρ′_{γ} .

Restricting this equation for α = 2, 3 to the the submanifold N yields

ρ_{1} ∧ ω_{2}|_{N} = ρ_{1} ∧ ω_{3}|_{N} = 0 .

This shows that ω_{2}|_{N} = ω_{3}|_{N} = 0, i.e. that J^{ϵ} ∈ Γ(N, Q) is parallel.

Proposition 21 The shape operator A of an ϵ-Kähler submanifold (N, J^{ϵ}, g_{N}) of a para-quaternionic Kähler manifold (M, g, Q) anticommutes with J := J^{ϵ}|_{T N}.

Proof: Let ξ be a normal vector field on N. Then the shape operator A^{ξ} ∈ Γ(End T N) is defined by

g(A^{ξ}X, Y) = −g(∇_{X}ξ, Y) = −g(∇_{Y} ξ, X) = g(ξ, ∇_{Y} X) .

Thus

g(A^{ξ}JX, Y) = g(ξ, ∇_{Y} (JX)) = g(ξ, J∇_{Y} X) = −g(Jξ, ∇_{Y} X) = −g(Jξ, ∇_{X}Y)
= g(ξ, J∇_{X}Y) = g(ξ, ∇_{X}(JY)) = g(A^{ξ}X, JY) = −g(JA^{ξ}X, Y) .

Corollary 7 Any ϵ-Kähler submanifold of a para-quaternionic Kähler manifold is minimal.

Proof: Since A^{ξ} anticommutes with J, we have A^{ξ} = −JA^{ξ}J^{−1}. Hence tr A^{ξ} = −tr A^{ξ} = 0.

8.2 Twistor construction of Kähler and para-Kähler submanifolds of para-quaternionic Kähler manifolds

Let (M, g, Q) be a para-quaternionic Kähler manifold and π_{Z} : Z^{ϵ} → M its ϵ-twistor space with the horizontal distribution ℋ_{Z}. For any ϵ-Kähler submanifold (N, J^{ϵ}, g_{N}) the section J^{ϵ} : N → Z^{ϵ} ⊂ Q defines an embedding of N into Z^{ϵ}. The image Ñ = J^{ϵ}(N) ⊂ Z^{ϵ} is called the canonical lift of N in the twistor space Z^{ϵ}. The following theorem gives the description of ϵ-Kähler submanifolds of M in terms of ϵ-complex horizontal submanifolds of Z^{ϵ}, i.e. submanifolds L ⊂ Z^{ϵ} such that 𝒥^{ϵ}T L = T L and T L ⊂ ℋ_{Z}.

Theorem 4 Let (N, J^{ϵ}, g_{N}) be an ϵ-Kähler submanifold of a para-quaternionic Kähler manifold (M, g, Q) and Ñ = J^{ϵ}(N) ⊂ Z^{ϵ} its canonical lift. Then

(i) Ñ ⊂ Z^{ϵ} is an ϵ-complex horizontal submanifold which is nondegenerate with respect to the canonical one-parameter family of metrics g_{t}^{ϵ} on Z^{ϵ}. Moreover, in the case ϵ = +1 the restriction of 𝒥^{ϵ} to Ñ is a para-complex structure in the strong sense.

(ii) Conversely, let L ⊂ Z^{ϵ} be an ϵ-complex horizontal submanifold which is nondegenerate with respect to g_{t}^{ϵ} and such that π_{Z}|_{L} : L → π_{Z}(L) ⊂ M is a diffeomorphism. Then its projection (N = π_{Z}(L), J^{ϵ}, g_{N}) is a (minimal) ϵ-Kähler submanifold of M, where

J^{ϵ} = dπ_{Z} ∘ 𝒥^{ϵ} ∘ (dπ_{Z})^{−1} : T N → T N , g_{N} = g|_{N} .

Proof: (i) Since J^{ϵ} is parallel, the submanifold Ñ = J^{ϵ}(N) ⊂ Z^{ϵ}, is horizontal. Its tangent bundle T Ñ ⊂ ℋ_{Z} is 𝒥^{ϵ}-invariant, since

dπ_{Z} ∘ 𝒥^{ϵ} = J^{ϵ} ∘ dπ_{Z} ,

on the horizontal distribution ℋ_{Z}, by the definition of 𝒥^{ϵ}, see (4.2). In the case ϵ = +1, J^{ϵ} is a para-complex structure in the strong sense, because J^{ϵ} is skew-symmetric for the metric g_{N}. Since (T_{z}Ñ, 𝒥_{z}^{ϵ}|Ñ) ≅ (T_{x}N, J_{x}^{ϵ}), x = π_{Z}(z), 𝒥^{ϵ} restricts to a para-complex structure in the strong sense on Ñ.

(ii) The ϵ-complex structure J^{ϵ} ∈ Γ(N, Q^{ϵ}) is parallel, since L = Ñ is horizontal. This proves that (N, J^{ϵ}, g_{N}) is an ϵ-Kähler submanifold of M.

Remark: The nondegeneracy assumption on the metric g_{t}^{ϵ}|L is essential even if we assume that dim L = 2n. Indeed there exist 2n-dimensional J^{ϵ}-invariant isotropic subspaces U ⊂ T_{x}M.

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