Homogeneous 8-manifolds admitting invariant Spin(7)-structures
Dmitri Alekseevsky, Ioannis Chrysikos, Anna Fino, Alberto Raffero
TL;DR
This paper classifies compact, simply connected homogeneous 8-manifolds with invariant Spin(7)-structures, providing explicit examples, their types, and analyzing the associated Spin(7)-connection with torsion.
Contribution
It offers a complete classification of such manifolds, explicit examples of invariant Spin(7)-structures, and analysis of their Spin(7)-connections with torsion.
Findings
Classification of all canonical presentations G/H with invariant Spin(7)-structures
Explicit examples of invariant Spin(7)-structures for each presentation
Description of the type of Spin(7)-structures and analysis of associated connections
Abstract
We study compact, simply connected, homogeneous 8-manifolds admitting invariant Spin(7)-structures, classifying all canonical presentations G/H of such spaces, with G simply connected. For each presentation, we exhibit explicit examples of invariant Spin(7)-structures and we describe their type, according to Fern\'andez classification. Finally, we analyse the associated Spin(7)-connection with torsion.
| canonical presentation | ||
|---|---|---|
| (1) | ||
| (2) | ||
| (3) | ||
| (4) | ||
| (5) | -full flag | |
| (6) | ||
| (7) | ||
| (8) |
| (i) | 10 | 2 | 2 | ||
|---|---|---|---|---|---|
| (ii) | 11 | 3 | 3 | ||
| (iii) | 12 | 2 | 4 | ||
| (iv) | 13 | 3 | 5 |
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Homogeneous 8-manifolds admitting invariant -structures
Dmitri Alekseevsky
Institute for Information Transmission Problems, B. Karetny per. 19, 127051, Moscow, Russia and Faculty of Science, University of Hradec Králové, Rokitanskeho 62, Hradec Králové 50003, Czech Republic
,
Ioannis Chrysikos
Faculty of Science, University of Hradec Králové, Rokitanskeho 62, Hradec Králové 50003, Czech Republic
,
Anna Fino
Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
and
Alberto Raffero
Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Abstract.
We study compact, simply connected, homogeneous 8-manifolds admitting invariant -structures, classifying all canonical presentations of such spaces, with simply connected. For each presentation, we exhibit explicit examples of invariant -structures and we describe their type, according to Fernández classification. Finally, we analyse the associated -connection with torsion.
1. Introduction
A -structure on an 8-manifold is characterized by the existence of a 4-form which can be pointwise written as
[TABLE]
for some basis of the cotangent space , where denotes the wedge product of covectors . Any such form is called admissible, and it gives rise to a Riemannian metric and an orientation on by the inclusion (cf. [16]). The existence of -structures depends on the topology of the manifold [21]. In particular, has to be orientable and spin.
By [12], the intrinsic torsion of a -structure can be identified with the exterior derivative of the corresponding 4-form . As a consequence, the Riemannian holonomy group is a subgroup of if and only if . In such a case, the metric is Ricci-flat, and the -structure is said to be torsion-free. More generally, the decomposition of the -module into irreducible submodules allows one to divide -structures into four classes, which were first described in [12]. Recently, a description of these classes in terms of spinorial equations has been obtained in [23].
Since is parallel with respect to the Levi Civita connection of if and only if , any other linear connection on preserving the -structure must necessarily have torsion. By [15], on there exists a unique connection with totally skew-symmetric torsion preserving both and a nontrivial spinor. It is given by .
The aim of this article is to study compact, simply connected 8-manifolds endowed with a -structure and acted on (almost) effectively and transitively by a compact connected Lie group of automorphisms. Every such manifold admits a presentation of the form , where is a closed subgroup of , and is a -invariant admissible 4-form. From the algebraic point of view, is a compact, simply connected, (almost) effective homogeneous space whose isotropy action on the tangent space is equivalent to the action of a closed subgroup of on .
Recall that every compact, simply connected, homogeneous space admits a canonical presentation, that is, it can be written as with a compact, connected, simply connected, semisimple Lie group and a closed connected subgroup of (see e.g. [8, 28]). All compact, simply connected, homogeneous 8-dimensional manifolds of a compact, connected, simply connected Lie group were classified in [18]. Moreover, a classification of compact simply connected Riemannian symmetric spin manifolds was given in [11]. A topological examination of these spaces allows us to obtain the following.
Theorem A**.**
- a)
The canonical presentations of all compact, simply connected, non-symmetric almost effective homogeneous spaces admitting a -structure are exhausted by
; 2. 2)
; 3. 3)
; 4. 4)
.
As smooth manifolds, the spaces a2) and a3) are diffeomorphic to , while the space a4) is diffeomorphic to . 2. b)
The compact, simply connected, symmetric spaces admitting a -structure are exhausted by , , , , and the exceptional Wolf space .
The manifold appearing in the above theorem is a torus bundle over the homogeneous Kähler-Einstein manifold , and hence a non-Kähler C-space. Invariant Einstein metrics on it were discussed in [8, 34]. When , is the direct product of with the total space of a circle bundle over . Furthermore, .
On the other hand, the homogeneous space is an example of a Calabi-Eckmann manifold. This is a torus bundle over , and hence also a non-Kähler C-space. By Jensen [13], the 5-sphere admits a unique invariant Einstein metric which coincides with the canonical metric. Consequently, the space admits a unique invariant Einstein metric.
Using general properties of symmetric spaces, we see that there are no invariant -structures on the manifolds described in part b) of Theorem A (cf. Lemma 3.2). Combining this with a case-by-case analysis of the homogeneous spaces appearing in part a) gives the following result.
Theorem B**.**
The canonical presentations of all compact, simply connected, almost effective homogeneous spaces admitting an invariant -structure are exhausted by , the infinite family , for , and the Calabi-Eckmann manifold .
It is remarkable that there are just a few examples of compact simply connected homogeneous spaces admitting invariant -structures. This is different from the case of -structures, where examples of this type are abundant (see [22, 31], and compare with the classification of compact almost effective homogeneous 7-manifolds given in [2]).
An example of invariant -structure inducing the bi-invariant metric on the homogeneous space was described in [12]. For the spaces , with , and , we describe the invariant Riemannian metrics and the invariant differential forms. This allows us to obtain a -parameter family of invariant -structures of mixed type on both of them. In particular, for both spaces we show that there exists an invariant -structure inducing the normal metric and whose associated -connection coincides with the canonical connection corresponding to the naturally reductive structure induced by . From this, it follows that has parallel torsion.
The paper is organized as follows. In Section 2, we recall some basic facts on -structures. Homogeneous 8-manifolds with an invariant -structure are discussed in Section 3. In Section 4, we review the main properties of simply connected homogeneous spaces. The main theorems A and B are proved in Sections 5 and 6, respectively. In particular, the infinite family is studied in Section 6.1, the Calabi-Eckman manifold is described in Section 6.2 and is analysed in Section 6.3. Explicit examples of invariant -structures on , with , and on the Calabi-Eckmann manifold are given is Section 7, where we also study the corresponding invariant -connection with torsion. Finally, in Appendix A we discuss the classification of all non-symmetric homogeneous presentations of .
We emphasize that the results of this paper are also useful to study compact 8-manifolds admitting other types of special structures, e.g. invariant -structures. This will be discussed in a forthcoming work.
2. Preliminaries
We begin recalling the main properties of 8-manifolds whose frame bundle admits a reduction to the Lie group . For more details, we refer the reader to [9, 15, 16, 21].
Consider the vector space , denote by the canonical basis, and by its dual basis. The group can be defined as the stabilizer in of the following 4-form on :
[TABLE]
is a compact, connected, simply connected Lie group of dimension 21. It is a subgroup of , as it preserves both the Euclidean inner product on and the volume form . Moreover, its center is and (see [9, Thm. 4] for a proof).
A -structure on a -dimensional manifold is a reduction of the structure group of its frame bundle from to . As is the stabilizer of the 4-form , such a reduction is characterized by the existence of a globally defined 4-form which can be pointwise identified with by means of an isomorphism . Any such form is called admissible, and it gives rise to a Riemannian metric and to an orientation on by the inclusion . We denote the Hodge operator associated with this metric and orientation by . Notice that is self-dual, i.e., . An explicit description of the metric in terms of the 4-form can be found, for instance, in [17, Sect. 4.3].
Remark 2.1**.**
By dimension counting, the -orbit of is not open in . Consequently, an admissible 4-form is not stable in the sense of Hitchin [14]. This differs significantly from the case of -structures on 7-manifolds, which are defined by stable 3-forms satisfying a suitable positivity condition. In eight dimensions, stability occurs for 3- and 5-forms, and the corresponding geometric structures are related to the group .
Since is both connected and simply connected, a connected 8-manifold admitting a -structure must be orientable and spin (with a preferred spin structure and orientation). These conditions are equivalent to the vanishing of the first and second Stiefel-Whitney classes of However, not every 8-dimensional Riemannian spin manifold admits -structures. More precisely, this is a topological issue, which can be characterized as follows.
Proposition 2.2** ([21]).**
An 8-dimensional orientable spin manifold admits -structures if and only if, for an appropriate choice of orientation, the following equation involving the Pontryagin classes , and the Euler characteristic of holds
[TABLE]
The intrinsic torsion of a -structure can be identified with the covariant derivative of the defining 4-form with respect to the Levi Civita connection of . When , the intrinsic torsion vanishes identically, the holonomy group of is a subgroup of and is Ricci-flat. In such a case, the -structure is said to be torsion-free or parallel. By [12], the intrinsic torsion can be also identified with the 5-form . Moreover, the -module splits into the direct sum of two irreducible submodules, say , with , . This allows one to divide -structures into four classes, which are completely characterized by . Besides the class of parallel -structures, corresponding to the condition , the following possibilities occur
class : balanced -structures, characterized by the condition ; 2.
class : locally conformal parallel -structures, characterized by the condition ; 3.
class : -structures of mixed type.
The 1-form is given by
[TABLE]
and it is called the Lee form of the -structure. In particular, is balanced if and only if , while whenever the -structure is locally conformal parallel (l.c.p. for short), see e.g. [17, Lemma 4.5.2].
Finally, according to [15, Thm. 1.1], any 8-dimensional manifold endowed with a -structure admits a unique metric connection with totally skew-symmetric torsion , satisfying . It is given by , where , and it is called the characteristic connection of .
3. Invariant -structures on homogeneous spaces
We now focus on homogeneous spaces. We start with the following definition.
Definition 3.1**.**
A -structure on an 8-dimensional manifold is called homogeneous or invariant if there exists a connected Lie group acting transitively and almost effectively on , preserving the 4-form .
In this case, is -equivariantly diffeomorphic to the homogeneous space , where is the (compact) stability group of a fixed point and is a -invariant 4-form on with pointwise stabilizer isomorphic to . Equivalently, the isotropy subgroup is a subgroup of , where denotes the isotropy representation of . Conversely, a homogeneous 8-manifold with admits invariant -structures.
As we are interested in compact examples, from now on we assume that is compact. Then is compact as well, and the Lie algebra of admits a reductive decomposition , where is the Lie algebra of and is an -invariant subspace of . Moreover, we can identify with the tangent space and the -invariant 4-form on with an -invariant 4-form on , which we shall denote by the same letter. Since the -action on is almost effective, the isotropy representation is injective, and we can identify the subalgebra with the isotropy subalgebra of the Lie algebra . Notice that the following constraints must hold
[TABLE]
The well-known interplay between - and -structures (see e.g. [10]) implies that every invariant -structure on a compact homogeneous 7-manifold gives rise to an invariant -structure on and, conversely, every invariant -structure on the 8-manifold induces an invariant -structure on . Consequently, in this case the complete list of homogeneous manifolds with an invariant -structure can be obtained from the results of [22, 31].
In the next sections, we will deal with the classification of the canonical presentations of compact, simply connected, almost effective homogeneous 8-manifolds that admit invariant -structures. The strategy to study this problem is the following. First, we consider all compact, simply connected, almost effective, homogeneous 8-manifolds with their canonical presentation, and we determine those satisfying the characterization of Proposition 2.2. This gives the list of the spaces admitting -structures. Then, for each space we investigate whether there exists an invariant admissible 4-form.
We conclude this section with some remarks.
Lemma 3.2**.**
A compact, simply connected, Riemannian symmetric space cannot admit any invariant -structure.
Proof.
Since all invariant differential forms on a Riemannian symmetric space are closed (see e.g. [35, p. 250]), any invariant -structure on a Riemannian symmetric space must be torsion-free. In particular, the corresponding invariant metric must be Ricci-flat. However, every compact simply connected Riemannian symmetric space is a direct product of irreducible symmetric spaces of compact type, which are Einstein with non-zero Einstein constant (cf. [6, 10.83]). ∎
More generally, since Ricci-flat homogeneous manifolds are flat [3], the class of compact connected homogeneous spaces admitting an invariant torsion-free -structure is exhausted by flat tori.
4. Simply connected 8-dimensional homogeneous spaces
Let be a compact, simply connected homogeneous space and let be a connected Lie group acting transitively and almost effectively on it. Starting from the corresponding presentation and using the results of [25, 28], it is always possible to obtain a presentation of the form , where is a compact, connected, simply connected, semisimple Lie group and is a connected closed subgroup (see e.g. [7, 8] for more details). This motivates the following.
Definition 4.1**.**
Let be a compact, simply connected homogeneous space. A canonical presentation of is a presentation of the form , where is a compact, connected, simply connected, semisimple Lie group and is a connected closed subgroup.
In what follows, we restrict our attention to such presentations.
Proposition 4.2**.**
([7])* For a compact, simply connected, homogeneous space with canonical presentation with semisimple, two possible cases occur:*
- (I)
; 2. (II)
.
In the first case, is the direct product of indecomposable homogeneous spaces which also satisfy (I), that is
[TABLE]
for some , with compact connected, simply connected and semisimple and closed, for all . Such homogeneous spaces are called prime.
In case (II), is the total space of a principal torus bundle over a product of prime homogeneous spaces. In particular, for any maximal torus in a compact complement of in , is the total space of the principal torus bundle
[TABLE]
where denotes . Note that the base space does not depend on the choice of and .
Example 4.3**.**
Consider the simply connected coset , where is the maximal compact connected subgroup of , and is a maximal torus. It is easy to see that this space satisfies condition (I). Hence, it is the direct product of the prime homogeneous spaces and .
Proposition 4.2 allows us to distinguish two classes of compact, simply connected homogeneous spaces. A particular example of (II) is the following (cf. [18, p. 80]).
Lemma 4.4** ([18]).**
Let be the total space of a -principal bundle over , with . If is simply connected, then it is diffeomorphic to .
Notice that for one obtains the circle bundle
[TABLE]
The space is a compact simply connected 5-dimensional spin manifold with , and hence diffeomorphic to the product of spheres . For , the space is diffeomorphic to , and it can be viewed as the unit tangent bundle of , see also [27, p. 6358]. For , we get the space
[TABLE]
where the embedding will be specified later (see Section 6.1). When , the space is a torus bundle over
[TABLE]
see Lemma 7.1 for a proof. Thus, as a manifold it is diffeomorphic to .
An inspection of the list of canonical presentations given in [8] (see also [4, Table 1]) combined with the results of [18, p. 81], allows us to obtain the canonical presentations for all compact, connected, simply connected, spin non-symmetric almost effective homogeneous 8-manifolds. They are described in Table 1.
More details on the cosets are given in Section 6, while further presentations of are discussed in Appendix A. Notice that the spaces and are all simply-connected homogeneous nearly Kähler manifolds.
5. Proof of Theorem A
In this section, we study the existence of -structures on the spaces appearing in Table 1 using the topological characterization of Proposition 2.2. Since all of these manifolds are orientable and spin, we only need to examine the constraint
[TABLE]
Recall that for a compact, connected, oriented 8-manifold , the following identity holds (cf. [32])
[TABLE]
where is the signature of namely the signature of the quadratic form associated to
[TABLE]
and is the fundamental homology class defined by the orientation. Moreover, the -genus of is given by
[TABLE]
Assume now that is also spin and let us denote by its spinor bundle and by the Dirac operator associated to a Riemannian metric on it. Then, decomposes as and one can consider the index of the (half) Dirac operator , which is given by . By the Atiyah-Singer Index Theorem, coincides with the -genus, i.e., . Moreover, if admits a metric of positive scalar curvature, then .
Let us now prove part a) of Theorem A.
Proposition 5.1**.**
Among the manifolds described in Table 1, only those appearing in the first four rows admit -structures.
Proof.
By [12, Sect. 7], we know that admits a -structure inducing the bi-invariant metric. Let be one of the manifolds of Table 1. As is a product of spheres with at least one of odd-dimension, it is parallelizable by [19] (see also [29]). Consequently, it admits -structures. Explicit examples of admissible 4-forms can be easily expressed in terms of a global coframe providing the absolute parallelism.
We now prove that the remaining spaces of Table 1 do not satisfy the relation (5.1). Indeed, apart from the full flag manifold , they are all products of the form , where is a 6-dimensional compact homogeneous nearly Kähler manifold. Therefore, it is easy to see that they satisfy , but their Euler characteristic is non-zero, since they are quotients of Lie groups of the same rank. For the same reason, the full flag manifold has . Moreover, , and by (5.2) and (5.3) we deduce that . Thus, none of these manifolds satisfies (5.1). ∎
We now prove part b) of Theorem A.
Proposition 5.2**.**
The 8-dimensional compact, simply connected, symmetric spaces admitting -structures are exhausted by the Lie group , the product of spheres and , the quaternionic projective space , the Grassmannian and the exceptional Wolf space .
Proof.
Since an 8-manifold admitting -structures is spin, we can focus on the list of compact simply connected spin symmetric spaces [11]. Up to a finite cover, we have to consider the following spaces
[TABLE]
Among these spaces, the half of them satisfy (5.1), namely , , , , and . For the first three of them, we have . As for the quaternionic projective space and the exceptional Wolf space, they both satisfy , and . Finally, the Grassmannian is such that , and , hence and . ∎
6. Proof of Theorem B
By Lemma 3.2, we know that all compact, simply connected, symmetric spaces cannot admit invariant -structures. Moreover, an explicit example of an invariant -structure inducing the bi-invariant metric on the homogeneous space is constructed in [12, Sect. 7]. Thus, we only need to consider the remaining spaces appearing in part a) of Theorem A, namely
[TABLE]
In order to simplify the presentation, we examine each case separately.
6.1. The infinite family
Let and
[TABLE]
Denote the Lie algebra of by and let be the Lie algebra of a maximal torus of The elements of can be viewed as complex block matrices of the form , with . Up to conjugation, any 1-dimensional subalgebra inside is described by a homomorphism
[TABLE]
The image of is the Lie algebra of a closed connected subgroup of if and only if . Moreover, using the Weyl group and the outer automorphisms of , it is always possible to reorder the elements of the triple in such a way that and and assume that all are integers. The stability algebra is and the Lie algebra is given by
[TABLE]
Since the stability group can be mapped by conjugation inside a maximal torus of , any coset space of the form is -equivariantly diffeomorphic to , where are integers such that and . Moreover, since we are interested in the simply connected case, we can assume that the triple consists of co-prime integers.
Denote by the bi-invariant metric on defined as and let be the orthogonal complement of inside with respect to . Then, and is a reductive decomposition. Thus, we can identify with the tangent space of at the identity coset . Let us consider the following orthogonal basis of :
[TABLE]
where
[TABLE]
, and . Notice that , for all and that . Moreover, and .
Proposition 6.1**.**
The space , with , and , admits -invariant -structures if and only if .
Proof.
Let us denote by the basis of given by the skew-symmetric matrices with in the -entry, in the -entry and zeroes elsewhere. The orthogonal transformation is given by , since
[TABLE]
Thus, the isotropy action of on yields the following subalgebra of :
[TABLE]
By using the basis of given in terms of the skew-symmetric matrices (cf. e.g. [5]), we see that a Cartan subalgebra of which occurs as the lift of a Cartan subalgebra of has the following expression:
[TABLE]
By comparing with , we see that is contained in if and only if . Consequently, admits invariant -structures if and only if . ∎
Explicit examples of invariant -structures on will be given in Section 7.1. In the next two examples, we describe some special spaces belonging to the family .
Example 6.2**.**
For and , the space coincides with the direct product , where we recall that (cf. [27]). Here, the 1-dimensional Lie subalgebra , where is a maximal torus of , corresponds to the diagonal embedding of in , and the existence of an invariant -structure follows from the inclusions .
Example 6.3**.**
Consider the direct product of the group manifold with the homogeneous space ,
[TABLE]
The stability algebra is given by , and the corresponding isotropy action is effective. In particular, acts trivially on the tangent space of , while sits diagonally inside and induces an irreducible representation when restricted to . Thus, this manifold belongs to the family for and , and we have the obvious diffeomorphisms
[TABLE]
The reductive decomposition of is
[TABLE]
where , and denotes the realification of the standard representation of on . Since the triple does not satisfy the condition , the space cannot admit any invariant -structure by Proposition 6.1.
6.2. The Calabi-Eckmann manifold
This space is the direct product of the homogeneous spaces and . The former is the canonical presentation of the unit 5-sphere acted on transitively and almost effectively by the Lie group with stability group at given by
[TABLE]
Now, the isotropy group lies inside . Recall that the latter is the stability subgroup of a unit vector in the -module and is the stability subgroup of a unit vector of the tautological -module (see e.g. [9, 31]). In particular, is a maximal subgroup of . Consequently, the homogeneous space admits invariant -structures. Explicit examples will be discussed in Section 7.2.
Remark 6.4**.**
The space is a Calabi-Eckmann manifold, i.e., a complex homogeneous non-Kähler manifold diffeomorphic to the product of two odd-dimensional spheres of dimension greater than two. In particular, it is a torus bundle over ,
[TABLE]
and, consequently, a C-space with (see [18]).
6.3. The space
For the sake of convenience, from now on we let
[TABLE]
Both manifolds and are Ledger-Obata spaces, i.e., of the form , with a compact simple Lie group and . Moreover, there is a natural isomorphism between the compact homogeneous space and the compact semisimple Lie group . Consequently, the corresponding 8-manifolds are diffeomorphic to . Notice however that does not belong to the family .
Since , the effective coset is covered by the almost effective simply connected coset . Thus, to conclude the proof of Theorem B, it is sufficient to show that the space does not admit any invariant -structure. To this aim, we will first describe the isotropy representation, and then the space of invariant forms.
Let and be the Lie algebras of and , respectively. A natural choice of an -invariant complement of in is given for instance by (see e.g. [26])
[TABLE]
and then . In our case, .
Consider on the bi-invariant metric , which is a multiple of the corresponding Killing form. The Lie algebra can be identified with the span of , and the matrices
[TABLE]
generate . By using the Gram-Schmidt process, we see that an orthogonal splitting of with respect to is given by
[TABLE]
where both and are irreducible. Therefore
[TABLE]
form a -orthonormal basis of such that and , respectively. Let us denote by the standard representation of . Then, one may identify and , where denotes another copy of .
Set now , and consider the Lie algebra
[TABLE]
This is a subalgebra of , since . Its defining representation is , where denotes the standard representation of on . Since sits inside the last summand of , sits inside and the pair induces the 8-dimensional effective homogeneous space . Summing up, a reductive decomposition of is
[TABLE]
where coincides with the tangent space to and with the tangent space to at the identity coset. Moreover, an orthonormal basis of is given by
[TABLE]
so that . The elements defined above, together with generate the stability algebra , which acts on via the isotropy representation as follows
[TABLE]
and
[TABLE]
We can now determine the invariant forms.
Lemma 6.5**.**
Let . Then, the following hold:
the space of -invariant 1-forms is trivial; 2. 2)
the space of -invariant 2-forms is 2-dimensional and it is generated by and ; 3. 3)
the space of -invariant 3-forms is 2-dimensional and it is generated by ; 4. 4)
the space of -invariant 4-forms 2-dimensional and it is generated by .
Proof.
The assertion for invariant 1-forms immediately follows from (6.3) and (6.4). For invariant 2-forms on , we have the equivariant decomposition
[TABLE]
Here, the first and the third module vanish, while from (6.3) we see that the second module is generated by . For the invariant 3-forms on , we see that only the modules and are non-trivial. In particular, the 9-dimensional spaces and do not contain any invariant element. Now, the claim for follows from the orthogonal decompositions
[TABLE]
∎
Proposition 6.6**.**
The space cannot admit any invariant -structure.
Proof.
We show that there are no invariant admissible 4-forms on this space. Let us consider the generic -invariant 4-form on
[TABLE]
where . By [17, Thm. 4.3.3], if defines a -structure inducing the orientation , then the norm of a vector with is proportional to the determinant of the matrix with the following entries
[TABLE]
Considering , an easy computation shows that all vanish. Thus, cannot define an invariant -structure. ∎
7. Explicit examples of invariant -structures
7.1. The infinite family
By Proposition 6.1, the space admits invariant -structures if and only if . In particular, since with , two different cases arise, namely
[TABLE]
For these two classes of homogeneous manifolds, the invariant objects (metrics, forms) are different.
We begin examining the family for , where and . In our computations, we shall keep on using the notation introduced in Section 6.1.
Let denote the orthogonal complement of inside the maximal torus with respect to . The space is spanned by and it is an abelian Lie algebra, i.e., . Thus, whenever , it generates a closed connected 2-dimensional abelian subgroup of , i.e., a 2-torus, which we denote by . This induces the 7-dimensional homogeneous space (cf. [27, 31])
[TABLE]
which is a circle bundle over .
Moving to the family , the reductive decomposition described before now reads
[TABLE]
where , and coincides with the 6-dimensional tangent space of . In the following, every , , will be viewed as an -module.
Remark 7.1**.**
When the simply connected coset is a non-Kähler C-space in the sense of Wang (cf. [1, 30]), i.e., a simply connected compact homogeneous complex manifold which is not Kähler. Indeed, since the integers are assumed to be relatively prime, the action of on
[TABLE]
is free. Consequently, the projection defines a principal torus bundle,
[TABLE]
and we have the following diagram:
[TABLE]
Since is a -bundle over the (full) flag manifold , it must be a C-space. Any complex structure on is automatically -invariant. Hence, whenever defines an -invariant complex structure on , the endomorphism defines an -invariant complex structure on .
In order to describe the set of -invariant metrics on , or equivalently, the space of -invariant inner products on , we first need to examine the properties of the modules .
Lemma 7.2**.**
Assume that . Then, the -modules and are pairwise inequivalent and irreducible. In contrast, decomposes into two irreducible 1-dimensional submodules, which we denote by and , respectively. These submodules are mutually equivalent, and they are not equivalent to . Consequently, the orthogonal -invariant decomposition
[TABLE]
is not in general unique.
Proof.
By (6.2), we see that the weights of the adjoint action of on are
[TABLE]
Since the stability group acts with different weights on , whenever , it follows that , , and are mutually inequivalent. Now, although is irreducible under the adjoint action of , it decomposes into two equivalent -invariant submodules, which are generated by and , respectively. Moreover, since , we see that . Hence, and can be replaced by any pair of orthogonal 1-dimensional submodules in . ∎
Fix some angle , consider the vectors
[TABLE]
and let , . Then, the -module is equivalent to , and the pair is an orthonormal basis of . Now, as a consequence of Lemma 7.2, we obtain the following.
Proposition 7.3**.**
Assume that . Let be the normal metric on induced by the bi-invariant metric on . Then, up to scaling, any -invariant metric on is given by
[TABLE]
for some positive real numbers . Thus, the space of -invariant metrics on , with , is 6-dimensional.
Let us now examine the spaces of invariant differential forms on . By the general theory of C-spaces described in [1], we can construct an equivariant isomorphism
[TABLE]
from which we deduce that . This also follows from the exact sequence (see [1, 18])
[TABLE]
In more geometric terms, any element , where denotes the centralizer of the stability subalgebra in , induces an invariant 1-form form and moreover an exact invariant 2-form on . Since is trivial, there is a bijection
[TABLE]
This means that , i.e., the dual 1-forms of induce -invariant 1-forms on . The same holds true for the dual 1-forms
[TABLE]
and their wedge product satisfies the relation . On the other hand, (6.2) gives
[TABLE]
and, similarly,
[TABLE]
Thus, a basis for the space of invariant 1-forms on , when , is given by
For the invariant 2-forms on , we obtain the following.
Lemma 7.4**.**
If the integers satisfy , then the space of -invariant 2-forms on is 4-dimensional and it is generated by the 2-forms .
Proof.
By the orthogonal decomposition , one obtains the orthogonal decomposition
[TABLE]
where It is easy to see that the 2-forms , , , and are -invariant. In particular
[TABLE]
Moreover, the -module vanishes if and only if the triple is non-zero. This follows from the relations
[TABLE]
For the remaining mixed terms corresponding to the modules , and , one similarly computes
[TABLE]
Using these results, we see that as long as and are non-zero, there are no further -invariant 2-forms. ∎
Remark 7.5**.**
Up to scaling, the 2-forms are the unique -invariant Kähler forms on the corresponding factors of the product . By Proposition 7.4, it follows that the 2-form is the fundamental 2-form associated to the invariant complex structure discussed in Remark 7.1, with . Of course, is not Kähler, since (cf. Appendix B).
We can now discuss the invariant 3-forms on . As we will see below, the condition appears in a natural way.
Lemma 7.6**.**
Assume that the integers , with , satisfy . Then, the space of -invariant 3-forms on is 8-dimensional and it is generated by the 3-forms
[TABLE]
and
[TABLE]
Proof.
Since , we obtain
[TABLE]
The 3-forms given by (7.3) are invariant, independently of the condition . They occur by considering the wedge product of the invariant 2-forms with the elements and , which span . Thus, they span the 6-dimensional factor . We mention that the invariant 3-forms and , for do not induce new forms. Now, for generic non-zero the third module vanishes, since
[TABLE]
Let us now consider the first factor. Due to the orthogonal decomposition , one has
[TABLE]
First, we show that for non-zero the first 6 modules vanish. Indeed, note that
[TABLE]
Computing the action of on the -forms appearing above, we see that for non-zero none of these 3-forms, or any linear combination of them, belong to the kernel of the isotropy action of . Let us finally prove that the condition is equivalent to
[TABLE]
This module is independent of the rotation that one may apply to . Moreover, and we obtain
[TABLE]
which shows that if and only if Finally, it is easy to see that these invariant 3-forms constitute a basis of if and only if . ∎
Combining Lemmas 7.4 and 7.6 allows us to describe the invariant 4-forms on .
Lemma 7.7**.**
Assume that the integers satisfy , with . Then, the space of -invariant 4-forms on is 10-dimensional, and it is generated by the 4-forms
[TABLE]
and
[TABLE]
where and are the invariant 3-forms described in (7.4). Note that and are obtained by applying the Hodge star operator of the bi-invariant metric to and , respectively.
Proof.
By the splitting , we obtain the -invariant orthogonal decomposition
[TABLE]
The module is generated by the invariant -forms and , which are obtained by wedging the 3-forms given in (7.3) with the generators of . Consider now the second summand . We can argue in a similar way as we did for . In detail, we know that splits into a direct sum of seven subspaces, and it is easy to see that any isomorphism does not contribute with further summands. An inspection of each subspace allows us to conclude that only one of them contains elements belonging to the kernel of the isotropy action when , namely . More precisely, we have
[TABLE]
This immediately follows from the identities
[TABLE]
To conclude the proof, we have to examine the first module in (7.5). It decomposes as follows:
[TABLE]
where the first three modules are 1-dimensional and they are generated by , , , respectively. A direct computation shows that the remaining modules are trivial. ∎
From the above proposition, we obtain the following.
Corollary 7.8**.**
Consider the homogeneous spaces with . Then, the 4-form
[TABLE]
is invariant if and only if . Whenever this condition is satisfied, induces an invariant -structure on .
Remark 7.9**.**
Due to the results of Appendix B, it is straightforward to check that there are no invariant closed 1-forms on as long as the integers satisfy . Thus, this homogeneous space cannot admit any invariant l.c.p. -structure.
We now describe a 5-parameter family of invariant -structures on inducing the most general invariant metric adapted to the reductive decomposition , with (cf. Proposition 7.3). The choice for the summand is just a matter of convenience, as it simplifies the computations afterwards.
Proposition 7.10**.**
The 4-form
[TABLE]
where , are positive real parameters, defines an invariant -structure on . It induces the metric , and its Lee form is given by
[TABLE]
where and are obtained by setting in the expressions of and , respectively. In particular, the -structure is always of mixed type.
Proof.
It is clear that the 4-form is invariant and admissible and that it induces the metric . For brevity, we denote by the corresponding Hodge operator. Recall that the Lee form is given by
[TABLE]
Using the results of Appendix B together with the definition of the Hodge operator, we obtain the expression of described above. To determine the Fernández type of this -structure, it is sufficient to observe that the Lee form is never closed (cf. Remark 7.9), and that the condition is equivalent to a system of two polynomial equations in the variables , which has no solutions under the constraints and , . Thus, the -structure is of mixed type. ∎
Corollary 7.11**.**
The invariant -structure defined by the admissible 4-form on , with , is of mixed type and its characteristic connection coincides with the canonical connection with respect to the naturally reductive structure induced by . In particular, its torsion form is parallel, i.e., .
Proof.
We already know that the 4-form defines a -structure of mixed type and it induces the normal metric . The homogeneous space is naturally reductive. Consequently, the canonical connection has totally skew-symmetric torsion (cf. [33]). Moreover, since and are -invariant, they are both parallel with respect to (see e.g. [20, Ch. X, Prop. 2.7]). Thus, the connection must coincide with the canonical connection by the uniqueness of the latter [15, Thm. 1.1], and the -invariant torsion form is -parallel.
Using the results of Appendix B, we can easily compute the expression of the torsion form, obtaining
[TABLE]
where and are obtained by setting in the expressions of and , respectively. ∎
To study the case , (cf. Example 6.2), we can proceed in a similar way. It turns out that the space of invariant 4-forms on has dimension 26 and that an invariant -structure on is given by
[TABLE]
This -structure is of mixed type with associated invariant metric and Lee form .
7.2. The Calabi-Eckmann manifold
Let and . is a direct product of prime homogeneous spaces, hence a reductive decomposition of the Lie algebra is given by
[TABLE]
Clearly, coincides with the tangent space of at the identity coset, and . Let us now consider the following basis of
[TABLE]
With this choice, the stability subalgebra is generated by the triple , and is a basis of , where the module is spanned by and Moreover, it is easy to check that this -basis is orthonormal with respect to the bi-invariant inner product
[TABLE]
From now on, we shall identify via the quaternionic metric on . A basis for the tangent space is given by the union of with a basis of . The latter may be chosen as follows
[TABLE]
so that Since and the action of on is trivial, we have
[TABLE]
Of course, is also invariant under the isotropy action, i.e., for any , while on we have
[TABLE]
Hence, in terms of skew-symmetric matrices we obtain the orthogonal transformations
[TABLE]
We are now ready to describe the space of -invariant forms on . As before, we shall denote by the dual of .
Lemma 7.12**.**
The spaces of -invariant -forms on , for have the following dimensions:
[TABLE]
A basis of each space is described in the proof.
Proof.
From the above description of the isotropy action, we immediately see that is spanned by .
Let us consider the space of invariant 2-forms. Since is an orthogonal -invariant decomposition, we obtain the invariant splitting
[TABLE]
where we identify . Obviously, the second module is spanned by , and the third one by . Thus we are left with
[TABLE]
The summand is easily seen to be trivial, while for the first summand we recall that
[TABLE]
Then, a direct computation shows that a basis of is given by .
Let us now consider the space of invariant 3-forms
[TABLE]
The last two summands are generated by and , respectively. The second summand is spanned by . Thus, it has dimension 9. Finally, we have
[TABLE]
where the first summand is trivial, while a basis of invariant 3-forms for the second summand is given by . Summing up, the space of invariant 3-forms on is 16-dimensional.
As for the -module , we have
[TABLE]
The second and the third module are both 9-dimensional. A basis of is given by
[TABLE]
and a basis of is
[TABLE]
The fourth module is 1-dimensional with generator . Also, , where spans the first summand while the second one is trivial. This finishes the proof. ∎
Using the basis of invariant 4-forms, we obtain the following.
Corollary 7.13**.**
The 4-form
[TABLE]
induces a -invariant -structure on .
Remark 7.14**.**
Using the Koszul formula, it is straightforward to check that there are no -invariant closed 1-forms on . Thus, we deduce that the Calabi-Eckmann manifold cannot admit any invariant l.c.p. -structure.
Now, we shall construct a family of invariant -structures inducing the general invariant metric on . To this aim, we first study the space of -invariant Riemannian metrics on , or equivalently, the space of -invariant inner products on the reductive complement . Recall that the decomposition of into inequivalent irreducible -modules is given by , with acting trivially on . Consequently, an -invariant inner product on must be of the form
[TABLE]
where is the inner product on given in (7.7), , and is an inner product on corresponding to a left-invariant Riemannian metric on . It follows from [24] that there exist suitable real parameters for which is isometrically isomorphic to , endowed with the left-invariant Riemannian metric induced by the following inner product on
[TABLE]
In particular, the canonical bi-invariant metric on corresponds to . The general invariant metric on is then given by the inner product
[TABLE]
In a similar way as we did for , it is now possible to show the following.
Proposition 7.15**.**
On the Calabi-Eckmann manifold , the invariant -structure given by
[TABLE]
induces the general invariant metric and it is of mixed type with Lee form
[TABLE]
Arguing as in the proof of Corollary 7.11, the next result immediately follows.
Corollary 7.16**.**
Let denote the invariant -structure on obtained by setting , for all . Then, the homogeneous space endowed with the metric induced by is naturally reductive, and its canonical connection coincides with the characteristic connection of . In particular, the torsion of is given by
[TABLE]
and it is -parallel.
Remark 7.17**.**
All examples of invariant -structures discussed in this section are of mixed type. Moreover, we observed that none of the homogeneous spaces considered here admits invariant l.c.p. -structures. It is an interesting open question to see whether there exist invariant balanced -structures on these homogeneous spaces.
Appendix A Presentations for not included in the family .
The -manifold has several different presentations as homogeneous space. For example, in Section 6.1 we analysed the pair , which induces the family . As a manifold, is diffeomorphic to and any homogeneous space of the form is equivariantly diffeomorphic to . We also discussed some special cases in Examples 6.2 and 6.3.
In this appendix, our goal is to focus on the presentations of which are not included in the family and which are still (almost) effective and simply connected.
Since the symmetric spaces and coincide, the isometry group of the symmetric Riemannian product is 15-dimensional. Consequently, we can focus on compact Lie algebras with smaller dimension. Moreover, as we are interested in invariant -structures, we restrict our attention to the case . In Table 2, we list all non-symmetric pairs satisfying the above constraints and which are different from .
Remark A.1**.**
In addition to the pairs considered in Table 2, one may also consider a fifth case with , i.e., . However, this corresponds to the stability algebra , which has rank bigger than 3.
To obtain the list appearing in Table 2, one has to consider pairs of the form
[TABLE]
with , , and such that the extra factor of sits diagonally inside the extra factor in (this always induces ). Although , here we use different indices for the summands and to emphasize that the abelian factor of does not coincide with the abelian factor of . The pairs appearing in Table 2 correspond to the following values of the parameters
[TABLE]
Let us examine Case (i) in detail. Here, the first factor of sits diagonally inside , where is a maximal torus of . In this case, the pair is almost effective and it induces the coset , which is simply connected. To see this, it is sufficient to prove that the space is simply connected, as is the product of this coset with the simply connected space . The principal circle bundle
[TABLE]
induces the exact sequence
[TABLE]
which reduces to . Since , the map must be trivial, whence . Thus, is simply connected.
Let and , so that . Consider the universal covering of and . Then, we obtain the equivariant diffeomorphism , from which it follows that , i.e., is covered by . Consequently, is covered by . More generally, we have the following.
Proposition A.2**.**
Any homogeneous manifold induced by a pair in Table 2, is covered by , which is diffeomorphic to .
Appendix B Details on the infinite family
Here, we collect some useful computational details related to the family , with coprime integers satisfying and . The notations used in this appendix are those introduced in Sections 6.1 and 7.1.
Let , where , and let be the -orthonormal basis of given in (6.1). The non-zero Lie brackets of the basis vectors are the following
[TABLE]
where , and . Moreover, the brackets , for are given in (6.2).
The differentials of the dual covectors of the basis vectors can be computed using the Koszul formula. In detail, for all we have
[TABLE]
whence we obtain
[TABLE]
Using these expressions together with the properties of the differential operator , it is possible to compute the exterior derivative of every invariant differential form on .
Acknowledgements: I.C. is grateful to Anton Galaev, Van Lê, Yurii Nikonorov and Henrik Winther for fruitful discussions related to the topic. He was supported by the project no. 19-14466Y of Czech Science Foundation (GAČR). D.A. was partially supported by Grant 18-00496S of the Czech Science Foundation. A.F. and A.R. were partially supported by GNSAGA of INdAM and by PRIN 2017 ”Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics.
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