Generalized almost even-Clifford manifolds and their twistor spaces


 Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.


Introduction
Generalized complex, quaternionic and exceptional geometries have attracted attention in mathematics and physics for the last two decades [1,7,8,14,16,20,24,26,27,29,30,34,39,43]. Pantilie [34] introduced the notion of generalized quaternionic manifold as well as the corresponding twistor space endowed with a connection-dependent generalized almost complex structure, whose integrability was discussed by Deschamps [17]. Recently, there has been some interest in Riemannian manifolds admitting even-Cli ord structures [2-5, 11-13, 15, 18, 22, 23, 25, 28, 31-33, 35-38, 41] and, in particular, Arizmendi and Had eld [5] constructed a twistor space for even-Cli ord manifolds. Motivated by these results, we de ne generalized even-Cli ord manifolds, which include generalized complex and quaternionic manifolds, and consider the construction of a twistor space endowed with a generalized complex structure by means of a suitable connection on the relevant principal bundle.
The paper is organized as follows. In Section 2, we recall standard material about generalized complex manifolds and Cli ord algebras. In Section 3, we introduce the notion of generalized even-Cli ord structure. In Section 4, we present the construction of the twistor space and its generalized almost complex structure. In Section 5, we prove the main theorem regarding the integrability of the generalized almost complex structure under certain conditions. endowed with the product of the Cli ord algebra. It is a Lie group and its Lie algebra is spin(n) = span{e i e j | ≤ i < j ≤ n}.
For future use, we reproduce the following lemma. Lemma 2.2. [5] Let A ∈ spin(r) ⊂ Cl r . Then, A = − if and only if there exist orthonormal vectors v , v ∈ R r such that A = v · v . In other words, the set of square roots of − ∈ Cl r is equal to the GrassmanianGr (R r ) of oriented -planes in R r .

2
Since every point ofGr (R r ) is identi ed with a bivector, we can consider Gr (R r ) ⊂ spin(r) ⊂ Cl r .

Generalized even-Cli ord structures . Generalized linear even-Cli ord structures
De nition 3.1. Let (e , . . . , er) be an orthonormal basis of R r with respect to the standard inner product, and V a real vector space.
• A generalized linear even-Cli ord structure of rank r on V, or generalized almost Cl r structure, is an algebra representation φ : such that each bivector e i e j , ≤ i < j ≤ r, is mapped to a linear complex structure φ(e i e j ) = J ij : V ⊕ V * −→ V ⊕ V * , R r . • Given that Cl r is a simple algebra with unit, the Lie algebra spin(r) is embedded by φ into End(V ⊕ V * ). • Let Spin(r) =: φ(Spin(r)) ⊂ End(V ⊕ V * ). The group that preserves the almost Cl r structure is the following normalizer subgroup N O(V⊕V * ) ( Spin(r)).
• A rank r generalized almost even-Cli ord structure on a smooth manifold M is a smoothly varying choice of a rank r generalized linear even-Cli ord structure on the generalized tangent space spaces of M. • A smooth manifold carrying a generalized almost even-Cli ord hermitian structure will be called an generalized almost even-Cli ord manifold, or a generalized almost Cl r -manifold for short.
This de nition can be rephrased as follows. There exists a (locally de ned) Riemannian vector bundle F → M of rank r such that the corresponding Cli ord bundle Cl (F) → M is globally de ned, and an injective algebra bundle homomorphism φ : is a generalized linear even-Cli ord structure. Examples.

A generalized almost
Cl -manifold is the same as a generalized almost complex manifold.

A generalized almost
Cl -manifold is the same as a generalized almost quaternionic manifold. 3. We know that almost complex manifolds and almost quaternionic manifolds provide examples supporting generalized complex and generalized quaternionic structures on their generalized tangent spaces respectively [20,34].
4. An almost even-Cli ord Hermitian manifold [4,31] is a Riemannian manifold with a (locally) de ned Riemannian vector bundle F and an algebra bundle map such that with respect to a local orthonormal frame (e , . . . , er) of F, the endomorphisms J ij = φ(e i e j ) are orthogonal and satisfy the Cli ord relations. Thus, an almost even-Cli ord Hermitian manifold admits a generalized almost even-Cli ord structure given by the local endomorphisms 5. The product of two hyperkähler manifolds admits four generalized almost Cl structures. Namely, the four possibilities listed in Example 3 of Subsection 3.1 using the sets of generalized complex structures on each factor induced by either the threre global complex structures or the three global symplectic structures. Note that, since they are given in blocks, every single one of these structures is integrable.
Let Q denote the sub-bundle of End(TM) whose bre at each point of M is the isomorphic image of spin(r). An almost Cl r -structure induces a reduction of structure to the subgroup where Spin(r) is the image of Spin(r) ⊂ Cl r in End(TM). Let P Gr be the principal bundle over M with such a reduced structure group. From now on, we x a connection on P Gr which, by de nition, is compatible with the canonical pairing (viewed as a metric) and the generalized almost Cl r structure. For instance, such a connection will induce a covariant derivative on sections of End(TM) which will make Q parallel.
Note that there exists a homomorphism

The twistor space construction
The construction of the twistor space for a generalized almost even-Cli ord manifold of arbitrary rank r emulates the one of Arizmendi and Had eld [5] for even-Cli ord structures.

. The bre: a real Grassmannian
Let r ≥ andG denote the real Grassmannian of oriented 2-planes in R r . This real Grassmannian has a natural complex structure since it is a Hermitian symmetric space. The complex structure I onGr (R r ) can be described using Cli ord multiplication as follows. Let S ∈Gr (R r ), which for a suitable frame (e , ..., er) of R r can be written as S = e ∧ e = e · e .
The tangent space toGr (R r ) at e e can be identi ed with (αs e · es + βs e · es) αs , βs ∈ R , and the almost complex structure I S is thus given by where v ∈ T SG r (R r ).

. The twistor space
Using the standard action ρr of Hr on the Hermitian symmetric spaceGr (R r ), we can associate the ber bundle π : Z ∼ = P Hr ×ρ rG r (R r ) −→ M which we will call the twistor space of the generalized almost Cl r -manifold M. In fact, for x ∈ M, the ber Zx can also be described as which is really φ(Gr (R r )), and we have Z ⊂ End(TM). Under this identi cation, φ becomes an inclusion map.

. The generalized tangent bundle of the twistor space
A given connection on P Gr induces connections on all its associated bundles. This applies, in particular, to P Hr and Z so that, at S ∈ Z, where V S = ker(dπ S ) is the tangent space to the ber at S ∈ Z, and H S is the corresponding horizontal subspace at S isomorphic to T π(S) M under dπ S . Such a decomposition implies Thus, the generalized tangent bundle TZ = TZ ⊕ T * Z splits as follows, Let us de ne We need to de ne a map dπ : is an isomorphism. It induces an isomorphism by pullback so that we can de ne, for any η ∈ V * S , dπ S (η) := and, for any α ∈ H * S , Thus, we have a connection dependent generalized projection map which restricts to an isomorphism dπ S | H S : H S −→ T π(S) M whose inverse, for notational sake, will generally be denoted by dπ − S .

De nition 4.1.
According to the decomposition TZ = H + H * + V + V * we will consider the following types of elds: • A eld X ∈ Γ(H) is called a horizontal vector eld.
• A horizontal vector eld X ∈ Γ(H) which is the horizontal lift of a vector eld on M is called basic.
• A di erential -form α which is the pullback under π of a -formᾱ on M will be called basic.
• A eld U ∈ Γ(V) will be called a vertical vector eld.
. The generalized almost complex structure on the twistor space Now, we will construct a generalized complex structure J on Z following the construction of almost complex structures on classical twistor spaces [5,9,10,40]. We need to de ne an automorphism Let S ∈ Z, X ∈ T S Z. Given the splitting let us write where the super-indices v and h denote vertical and horizontal parts of the generalized tangent bundle, i.e.
First, let us extend the complex structure I of the bre to TZ and denote it by the same letter, i.e. for

Lemma 4.1. Let X, Y be basic vector elds, α be a horizontal di erential -form, η a vertical -form, U, W vertical vector elds and f
Proof. Part (a) follows from [42,Prop. 8.2] since the elds on Z are π-related to the elds on M. Part (b) follows from (a) since U is π-related to . Parts (c) -(g) follow from Cartan's formula.
2 From now on, given any vector eld W, we will denote its local ow by ϕ W t . Note that the ow ϕ X t of a basic vector eld preserves the bration structure of Z and, since each point S ∈ Z π(S) is itself the complex structure of T S Z π(S) , ϕ X t also sends the complex structure S of T S Z π(S) to the complex structure ϕ X t (S) of T ϕ X t (S) Z π(S) .

Lemma 4.2. Let X be a basic vector eld. Then
2

Lemma 4.3. Let X be a basic vector eld and η a vertical di erential -form. Then
Proof. Let Y be a basic vector eld and U a vertical vector eld. Consider
2 Lemma 4.4. Let ξ be vertical di erential -form, X, Y horizontal vector elds and β a horizontal di erential -form. Then . . , αn} be a local frame of H consisting of local basic vectors and forms such that where we used Lemma 4.1(g).

Lemma 4.5. Let R be the curvature tenrsor of the induced connection on
where X and Y are any local horizontal extensions of X S and Y S .
Proof. This follows from [42, Thm. 3.2, p. 118] applied to the bundle End(TM) with its induced connection as an associated bundle of P Gr .

Integrability of J
Let S ∈ Z, x = π(S ) ∈ M andX,Ȳ,Z ∈ Γ(TM). We will consider the following two conditions on the curvature and generalized torsion of the connection on TM: Theorem 5.1. Let M be a generalized almost even-Cli ord manifold endowed with a connection on its reduced structure bundle P Gr , Z its twistor space and J its connection-dependent generalized almost complex structure. If the curvature and the generalized torsion satisfy conditions (1) and (2) respectively, then J is integrable, Remark. Note that condition (1) is not invariant under a change of sign of the complex structure on the ber.
We will split the proof of the theorem in the following three lemmas.

Lemma 5.3. If the curvature and the generalized torsion satisfy conditions
Before proving the theorem, let us consider the following example.

Example.
A manifold with a parallel even-Cli ord structure is a Riemannian manifold M such that the almost even-Cli ord structure is parallel with respect to the Levi-Civita connection [31]. The even-Cli ord structure {J ij } gives rise to a generalized almost even-Cli ord structure Since the curvature tensor and the local generalized almost complex structures are diagonal on TM ⊕ T * M, condition (1) reduces to the curvature condition for the integrability of the complex structure on the classical twistor space of M (in the notation of [5]) which parallel even Cli ord manifolds satisfy [5,Lemma 3.2]. Since the Levi-Civita connection is torsion free, the generalized torsion of the induced connection on TM is zero, and condition (2) is also ful lled. Thus, the twistor space of the induced generalized almost even-Cli ord structure of a parallel even-Cli ord manifold admits an integrable generalized complex structure, which is consistent with the integrability of the complex structure of its classical twistor space.

. Proof of Lemma 5.2
Let We will examine one term at a time. The rst term is The second term is The third term is The fourth term is Using Lemma 4.3 Let us write (J(X + α))v = Indeed, we have  Consider where we have used that φ is a bre-wise algebra homomorphism.
By Lemma 4.1, L U (J(X + α)) f is horizontal. Hence, let us consider A a basic vector eld,Ā its projection to M and Thus, where we have used that φ is a bre-wise algebra homomorphism.
For a vertical form ξ and by Lemma 4.4 Thus, by Lemma 4.5, Thus, if the curvature satis es (1) 2

. Proof of Lemma 5.3
Let X, Y be basic vector elds, i.e. horizontal lifts of vectorsX,Ȳ on M, and α, β be pullbacks of 1-formsᾱ,β on M. We will prove the following two lemmas. (2), then

Lemma 5.4. If the generalized torsion T satis es condition
Lemma 5.5. If the curvature R satis es condition (1), then .

. Proof of Lemma 5.4
In order to carry out the calculations, let us consider • a curve γ(t) in the image ofS(x) such that γ( ) = S andγ( ) = Y S (it exists due to the tangency to H S ). Clearly, the curveγ(t) = π(γ(t)) in M is such thatγ( ) = x ,γ( ) =Ȳx and γ(t) =S(γ(t)); • a curve δ(t) in the image ofS(x) such that δ( ) = S andγ( ) = A S (it exists due to the tangency to H S ).
We will actually prove the following.

Proof. Recall that
We will examine one term at a time.

Proof. We have
Indeed, Since which concludes the proof of Claim 2 and Lemma 5.7.
Proof. We will only prove the rst identity since the second one is analogous. Consider .
Proof. We have  Claim 3.