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# Hypercomplex Structures Associated To Quaternionic Manifolds

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*in*Differential Geometry and its Applications 9(3):273-292 · December 1998

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Abstract

If M is a quaternionic manifold and P is an S 1 -instanton over M , then Joyce constructed a hypercomplex manifold we call P(M ) over M . These hypercomplex manifolds admit a U(2)-action of a special type permuting the complex structures. We show that up to double covers, all such hypercomplex manifolds arise in this way. Examples, including that of a hypercomplex structure on SU(3), show the necessity of including double covers of P(M ). 1. Introduction In [13], it was shown that over any quaternionic Kahler manifold there is a hyperK ahler manifold with an isometric action of SO(3) rotating the complex structures. The hyperKahler manifolds which arise this way were characterised by the properties of this action and it was shown how to recover the original quaternionic Kahler manifold via a moment map construction. The purpose of this article is to study similar constructions and results for quaternionic and hypercomplex structures---geometries that do not involve Riemannian metr...

- ... is the (complex) projectivization of H * 0 . To describe the complex structure on Z it will be convenient to describe first the complex structure on the total space of H * 0 with the zero section removed (in fact this space carries not only complex structure, but a hypercomplex structure, see [25]). Let us choose a torsion free G-connection ∇. ...... Thus we got an almost complex structure on H * 0 . It is was shown in [25], Theorem 3.2, that it is integrable and is independent of a choice of a torsion free connection ∇. Now the non-zero complex numbers C * act holomorphically by the product on H * 0 \{0}, and the quotient is equal to the twistor space Z = P(H * 0 ). ...Article
- Oct 2010
- J GEOM PHYS

On any quaternionic manifold of dimension greater than 4 a class of plurisubharmonic functions (or, rather, sections of an appropriate line bundle) is introduced. Then a Monge-Amp\`ere operator is defined. It is shown that it satisfies a version of theorems of A. D. Alexandrov and Chern-Levine-Nirenberg. These notions and results were previously known in the special case of hypercomplex manifolds. One of the new technical aspects of the present paper is the systematic use of the Baston differential operators, for which we prove a new multiplicativity property. - ... In this paper, we will formulate and prove the 1-to-1 correspondence 2 (locally) between conformal hypercomplex manifolds of quaternionic dimension n H + 1 and quaternionic manifolds of dimension n H . Furthermore, we show that this 1-to-1 correspondence is also applicable between the subset of hypercomplex manifolds that are hyper-Kähler and the subset of quaternionic manifolds that are quaternionic-Kähler. In the mathematics literature the map between quaternionic-Kähler and hyper-Kähler manifolds is constructed by Swann [21], and its generalization to quaternionic manifolds is treated in [22] . Here, we give explicit expressions for the complex structures and connections that are needed to apply these results in the context of the conformal tensor calculus in supergravity. ...... It would be very interesting to understand the role of general quaternionic manifolds in the context of such compactifications. In mathematics, the map that is described in this paper was investigated in [21, 22]. We have pointed out that the corresponding manifolds in the hypercomplex/hyper-Kähler picture are conformal, and discovered some new properties. ...We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by '\xi-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Kaehler manifolds is mapped to quaternionic-Kaehler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other. Comment: 54 pages, 2 figures; v2: small corrections, version to be published in CMP; v3: changes of statement on (3.5)
- ... In this paper, we will formulate and prove the 1-to-1 correspondence 2 (locally) between conformal hypercomplex manifolds of quaternionic dimension n H + 1 and quaternionic manifolds of dimension n H . Furthermore, we show that this 1-to-1 correspondence is also applicable between the subset of hypercomplex manifolds that are hyper-Kähler and the subset of quaternionic manifolds that are quaternionic-Kähler. In the mathematics literature the map between quaternionic-Kähler and hyper-Kähler manifolds is constructed by Swann [21], and its generalization to quaternionic manifolds is treated in [22] . Here, we give explicit expressions for the complex structures and connections that are needed to apply these results in the context of the conformal tensor calculus in supergravity. ...... It would be very interesting to understand the role of general quaternionic manifolds in the context of such compactifications. In mathematics, the map that is described in this paper was investigated in [21, 22]. We have pointed out that the corresponding manifolds in the hypercomplex/hyper-Kähler picture are conformal, and discovered some new properties. ...We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by `ξ-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Khler manifolds is mapped to quaternionic-Khler manifolds. We relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other.
- ... We recall the notion of a Swann bundle [43, 34, 36]. On a 4n dimensional QKT manifold M it is defined by U(M ) = (P × Sp(n)Sp(1) H * )/{±}, where H * are the nonzero quaternions. ...... On a 4n dimensional QKT manifold M it is defined by U(M ) = (P × Sp(n)Sp(1) H * )/{±}, where H * are the nonzero quaternions. It carries a hypercomplex structure [43, 34, 36]. If M is of instanton type and the condition ...ArticleFull-text available
- Jan 2002

The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor, thus connecting two structures arising from quantum field theories and supersymmetric sigma models with Wess-Zumino term. - ... The analogous result for a para-quaternionic submanifold of a para-quaternionic Kähler manifold was proved in [27]. In fact, by arguing as in [6], [24], a more general result can be proved. ...... Proof. For quaternionic case the proof was given in [6], [24]. For paraquaternionic case we observe that the proof in [6] is directly adaptable. ...Article
- Jan 2009

After a concise introduction on (para-)quaternionic geometry, we report on some recent results concerning para-quaternionic Hermitian and Kähler manifolds and their special submanifolds. The second part of the paper is devoted to treat in a unified way some basic matters on (para-)complex submanifolds of (para-)quaternionic manifolds. - ... It has been shown in [164] that this can be implemented in superconformal tensor calculus to construct the actions of hypermultiplets in any quaternionic- Kähler manifold from a hyperkähler cone. Similarly, it has been proven in [184, 185] that any quaternionic manifold is related to a hypercomplex manifold. Locally there is a vielbein f iA X (with i = 1, 2 and A = 1, . . . ...Thesis
- Oct 2003

Sinds de geboorte van de deeltjesfysica, na de ontdekking van het elektron door Thomson in 1897, is er een enorme vooruitgang geboekt in de beschrijving van waarneembare processen in de natuur. Om het gedrag van deeltjes te kunnen verklaren op (sub)atomaire schaal, werd omstreeks 1920 de quantummechanica ontwikkeld. Men realiseerde zich, afgaand op de uitkomsten van experimenten, dat alle deeltjes een fundamentele eigenschap bezitten: genaamd 'spin'. .... Zie: Samenvatting - ... The bi-invariant metric on SU(3) realises the hypercomplex structure as a strong HKT manifold whose torsion-three form c is given by (5.2) [27]. The symmetry group of this HKT structure is precisely H × K = SU(3) × U(1) and the map ν realises SU(3) as a twisted associated bundle over CP(2) [38]. ...Article
- Mar 2012
- ADV MATH

We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra Betti numbers are zero. We show that these form a special class of solvable Lie groups and provide a structural characterisation. We provide many examples of multi-moment maps for different geometries and use them to describe manifolds with holonomy contained in G_2 preserved by a two-torus symmetry in terms of tri-symplectic geometry of four-manifolds. - ... He then goes on to consider hypercomplex structures on more advanced structures derived from instantons, the most notable of which is the so-called Twisted Swann Bundle V P (M ) = P × S 1 V(M ) associated to an S 1 -instanton P on M . This work is extended even farther in [60]. Then, of course, there's the indescribable amount of literature produced by authors Boyer, Galicki, and Mann. ...Technical ReportFull-text available
- Jan 2014

- ... In this paper, we explain the 1-to-1 correspondence 1 (locally) between conformal hypercomplex manifolds of quaternionic dimension n H +1 and quaternionic manifolds of dimension n H . Furthermore, we show that this 1-to-1 correspondence is also applicable between the subset of hypercomplex manifolds that are hyper-Kähler and the subset of quaternionic manifolds that are quaternionic-Kähler. The map between quaternionic-Kähler and hyper-Kähler manifolds is constructed by Swann [6], and its generalization to quaternionic manifolds is treated in [7]. Here, we give explicit expressions for the complex structures and connections, curvatures and symmetries. ...Article
- Jan 2006
- INT J GEOM METHODS M

We review the map between hypercomplex manifolds that admit a closed homothetic Killing vector (i.e. `conformal hypercomplex' manifolds) and quaternionic manifolds of 1 dimension less. This map is related to a method for constructing supergravity theories using superconformal techniques. An explicit relation between the structure of these manifolds is presented, including curvatures and symmetries. An important role is played by `\xi transformations', relating connections on quaternionic manifolds, and a new type `\hat\xi transformations' relating complex structures on conformal hypercomplex manifolds. In this map, the subclass of conformal hyper-Kaehler manifolds is mapped to quaternionic-Kaehler manifolds. - ... Accordingly, a quaternionic map [4] between quaternionic manifolds is, essentially, a map admitting a holomorphic lift between the corresponding twistor spaces. This generalizes the well-known notion of quaternionic submanifold (necessarily, totally geodesic with respect to any compatible torsion free connection [1] ; see [9] ). However, not all submanifolds of a quaternionic manifold are quaternionic (take, for example, any hypersurface). ...Article
- Aug 2011
- Adv Geom

We introduce the notion of CR quaternionic map and we prove that any such real-analytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold $M$, of dimension $4k-1$, of a quaternionic manifold $N$, such that $TM$ generates a quaternionic subbundle of $TN|_M$, of (real) rank $4k$, there exists, locally, a quaternionic submanifold of $N$, containing $M$ as a hypersurface. - ... It has been shown in [30] that this can be implemented in superconformal tensor calculus to construct the actions of hypermultiplets in any quaternionic-Kähler manifold from a hyperkähler cone. Similarly, it has been proven in [88, 89] that any quaternionic manifold is related to a hypercomplex manifold. Locally there is a vielbein f iA X (with i = 1, 2 and A = 1, . . . ...We investigate N = 2, D = 5 supersymmetry and matter-coupled supergravity theories in a superconformal context. In a first stage we do not require the existence of a lagrangian. Under this assumption, we already find at the level of rigid supersymmetry, i.e. before coupling to conformal supergravity, more general matter couplings than have been considered in the literature. For instance, we construct new vector-tensor multiplet couplings, theories with an odd number of tensor multiplets, and hypermultiplets whose scalar manifold geometry is not hyperkähler. Next, we construct rigid superconformal lagrangians. This requires some extra ingredients that are not available for all dynamical systems. However, for the generalizations with tensor multiplets mentioned above, we find corresponding new actions and scalar potentials. Finally, we extend the supersymmetry to local superconformal symmetry, making use of the Weyl multiplet. Throughout the paper, we will indicate the various geometrical concepts that arise, and as an application we compute the non-vanishing components of the Ricci tensor of hypercomplex group manifolds. Our results can be used as a starting point to obtain more general matter-couplings to Poincaré supergravity.
- ... Then, at least locally, we have T C M = H ⊗ W , where H and W are complex vector bundles of rank 2 and 2k , respectively, and the structural group of H is SL(2, C) (H and W exist globally if and only if the vector bundle generated by the admissible linear complex structures on M is spin). Denote H = (L * ) k/k+1 ⊗ H. Then H \ 0 is endowed with a natural hyper-complex structure ([10] ; see [9] ), such that the projection onto M is twistorial. In particular, on endowing H \ 0 with one of the admissible complex structures (corresponding to some imaginary quaternion of length 1 ) then H \ 0 is the total space of a holomorphic principal bundle over the twistor space Z of M , with group C \ {0} . ...Article
- May 2013

We characterise the integrability of any co-CR quaternionic structure in terms of the curvature and a generalized torsion of the connection. Also, we apply this result to obtain, for example, the following. (1) New co-CR quaternionic structures built on vector bundles over a quaternionic manifold M, whose twistor spaces are holomorphic vector bundles over the twistor space Z of M. Moreover, all the holomorphic vector bundles over Z, which are positive and isotypic when restricted to the twistor lines, are obtained this way. (2) Under generic dimensional conditions, any manifold endowed with an almost f-quaternionic structure and a compatible torsion free connection is, locally, a product of a hypercomplex manifold with some power of the space of imaginary quaternions. - ... It follows that, we may assume that there exists an antiholomorphic anti-involution τ : L → L covering σ. Note that, if we restrict L to the real twistor spheres, then τ give antiholomorphic anti-involutions as in Section 1. Also, for classical quaternionic manifolds of dimensions at least eight it is known that L 2 exists globally [17] , [13] . ...Article
- Nov 2014

We describe the Penrose transform for the `quaternionic objects' whose twistor spaces are complex manifolds endowed with locally complete families of embedded Riemann spheres with positive normal bundles. - ... The following result was proved in [2], see also [17]. ...ArticleFull-text available
- Jan 2

It is a report on some recent results concerning the almost com-plex submanifolds of a quaternionic, in particular quaternionic Kähler, mani-fold. Some extensions of these results to submanifolds of a quaternionic Kähler manifold with torsion (QKT manifold) are considered. - The geometry arising from Michelson & Strominger's study of N=4B supersymmetric quantum mechanics with superconformal D(2, 1; alpha)-symmetry is a hyperKahler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters alpha are related via changes in potentials for the HKT target spaces. For alphanot equal0, -1, we describe how each such HKT manifold M-4m is derived from a space N4m-4 which is quaternionic Kahler with torsion and carries an Abelian instanton.
- Article
- Mar 2006

The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a `projective cone', a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each possible holonomy algebra. - Article
- Dec 2005

This is part two of a series of four methodological papers on (bi)quaternions and their use in theoretical and mathematical physics: 1- Alphabetical bibliography, 2- Analytical bibliography, 3- Notations and terminology, and 4- Formulas and identities. This quaternion bibliography will be further updated and corrected if necessary by the authors, who welcome any comment and reference that is not contained within the list. - Classification results are given for (i) compact quaternionic K\"ahler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperK\"ahler manifolds with a cohomogeneity-two action of a semi-simple group preserving each complex structure, (iii) compact 3-Sasakian manifolds which are cohomogeneity one with respect to a group of 3-Sasakian symmetries. Information is also obtained about non-compact quaternionic K\"ahler manifolds of cohomogeneity one and the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras.
- Article
- Nov 2005

This is part one of a series of four methodological papers on (bi)quaternions and their use in theoretical and mathematical physics: 1- Alphabetical bibliography, 2- Analytical bibliography, 3- Notations and terminology, and 4- Formulas and identities. This quaternion bibliography will be further updated and corrected if necessary by the authors, who welcome any comment and reference that is not contained within the list. - Article
- Sep 1999
- MEM AM MATH SOC

Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map \Pi : \wedge^2 W \to V. If the skew symmetric vector valued bilinear form \Pi is nondegenerate then (M,Q) is endowed with a canonical pseudo-Riemannian metric g such that (M,Q,g) is a homogeneous quaternionic pseudo-K\"ahler manifold. The construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any spin(V)-equivariant linear map \Pi : Sym^2 W \to V a homogeneous quaternionic supermanifold (M,Q) is constructed and, moreover, a homogeneous quaternionic pseudo-K\"ahler supermanifold (M,Q,g) if the symmetric vector valued bilinear form \Pi is nondegenerate. - Article
- Feb 2004
- J REINE ANGEW MATH

The target space of a (4, 0) supersymmetric two-dimensional sigma model with a Wess-Zumino term has a connection with a totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT and HKT manifolds and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry, if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor. In this way we connect two structures arising from quantum field theories and supersymmetric sigma models with Wess-Zumino term. - In [KSW97a] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kahler manifolds. In the present article we show that the only manifolds in the limit case, i. e. the only manifolds where the lower bound is attained as an eigenvalue, are the quaternionic projective spaces. We use the equivalent formulation in terms of the quaternionic Killing equation introduced in [KSW97b] and show that a nontrivial solution defines a parallel spinor on the associated hyperkahler manifold. AMS Subject Classification: 53C25, 58G25 Contents 1 Introduction 2 2 Semiquaternionic Vector Spaces and Representations 3 3 Principal Bundles on Quaternionic Kahler Manifolds 5 4 The Levi--Civit'a Connection on S and c M 8 5 Quaternionic Killing Spinors 10 6 The Geometry of c M and Application to Spinors 13 6.1 The Hyperkahler Structure of c M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2 Reinterpretation of the Quaternionic Killing Equation . . . . . . . . ....
- Article
- Apr 2011
- INT MATH RES NOTICES

A hypercomplex structure on a smooth manifold is a triple of integrable almost complex structures satisfying quaternionic relations. The Obata connection is the unique torsion-free connection that preserves each of the complex structures. The holonomy group of the Obata connection is contained in $GL(n, \mathbb{H})$. There is a well-known construction of hypercomplex structures on Lie groups due to Joyce. In this paper we show that the holonomy of the Obata connection on SU(3) coincides with $GL(2, \mathbb{H})$. - Article
- Apr 2008
- ANN GLOB ANAL GEOM

The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a ‘projective cone’, a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each possible holonomy algebra. - The internal space of a N = 4 supersymmetric model with Wess–Zumino term has a connection with totally skew-symmetric torsion and holonomy in SP(n). We study the mathematical background of this type of connection. In particular, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.
- We review some recent observations in theories with eight supercharges. We point out that such theories can be generalized by starting from the equations of motion rather than from an action. We show that vector multiplets constructed in such a way can have more general Fayet-Iliopoulos terms and that the scalar fields of hypermultiplets can be coordinate functions on more general target spaces. Although our discussion holds in five dimensions, the results can easily be extended to other dimensions.
- Article
- Jan 2009

After a concise introduction on (para-)quaternionic geometry, we report on some recent results concerning para-quaternionic Hermitian and Kahler manifolds and their special submanifolds. The second part of the paper is devoted to treat in a unified way some basic matters on (para-)complex submanifolds of (para-)quaternionic manifolds. - Article
- Jan 1999
- J REINE ANGEW MATH

We study deformations of hypercomplex structures on compact homogeneous spaces through the complex deformation theory of the associated twistor spaces. In general, we find complete parameter spaces of hypercomplex structures associated to compact semi-simple Lie groups. In particular, we discover the moduli space of hypercomplex structures on products of Hopf surfaces and on compact associated bundles of quaternionic projective spaces. - The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor, thus connecting two structures arising from quantum field theories and supersymmetric sigma models with Wess-Zumino term.
- Book
- Jan 2008

Sasakian manifolds were first introduced in 1962. This book's main focus is on the intricate relationship between Sasakian and Kähler geometries, especially when the Kähler structure is that of an algebraic variety. The book is divided into three parts. The first five chapters carefully prepare the stage for the proper introduction of the subject. After a brief discussion of G-structures, the reader is introduced to the theory of Riemannian foliations. A concise review of complex and Kähler geometry precedes a fairly detailed treatment of compact complex Kähler orbifolds. A discussion of the existence and obstruction theory of Kähler-Einstein metrics (Monge-Ampère problem) on complex compact orbifolds follows. The second part gives a careful discussion of contact structures in the Riemannian setting. Compact quasi-regular Sasakian manifolds emerge here as algebraic objects: they are orbifold circle bundles over compact projective algebraic orbifolds. After a discussion of symmetries of Sasakian manifolds in Chapter 8, the book looks at Sasakian structures on links of isolated hypersurface singularities in Chapter 9. What follows is a study of compact Sasakian manifolds in dimensions three and five focusing on the important notion of positivity. The latter is crucial in understanding the existence of Sasaki-Einstein and 3-Sasakian metrics, which are studied in Chapters 11 and 13. Chapter 12 gives a fairly brief description of quaternionic geometry which is a prerequisite for Chapter 13. The study of Sasaki-Einstein geometry was the original motivation for the book. The final chapter on Killing spinors discusses the properties of Sasaki-Einstein manifolds, which allow them to play an important role as certain models in the supersymmetric field theories of theoretical physics. - Article
- Nov 2014
- TOHOKU MATH J

Let $M$ be a quaternionic manifold, $\dim M=4k$, whose twistor space is a Fano manifold. We prove the following: (a) $M$ admits a reduction to $Sp(1) \times GL(k,H)$ if and only if $M=HP^k$, (b) either $b_2(M)=0$ or $M=Gr_2(k+2,C)$. This generalizes results of S. Salamon and C.R. LeBrun, respectively, who obtained the same conclusions under the assumption that $M$ is a complete quaternionic-Kaehler manifold with positive scalar curvature. - We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) A map between quaternionic manifolds endowed with the nonintegrable almost twistorial structures is twistorial if and only if it is quaternionic and totally-geodesic. As an application, we describe the quaternionic maps between open sets of quaternionic projective spaces.

- Foliation reduction and self-duality
- J.F. Glazebrook
- F.W. Kamber
- H. Pedersen
- A. Swann
- J.F. Glazebrook
- F.W. Kamber
- H. Pedersen
- A. Swann

- ArticleFull-text available
- Jan 1965

- Article
- Jan 1975
- ANN MAT PUR APPL

Quaternion generalized fiber bundles [(E)\tilde]n\mathbbQ ® X\tilde E_n^\mathbb{Q} \to X are studied, both isomorphic to global tensorial product En\mathbbQ Ä\mathbbQ E1\mathbbQ (En\mathbbQ , E1\mathbbQE_n^\mathbb{Q} \otimes _\mathbb{Q} E_1^\mathbb{Q} (E_n^\mathbb{Q} , E_1^\mathbb{Q} ordinary quaternion fiber bundles right and left respectively) and quite general ones. A cohomology class e([(E)\tilde]n\mathbbQ ) Î H2 (X;\mathbbZ2 )\varepsilon (\tilde E_n^\mathbb{Q} ) \in H^2 (X;\mathbb{Z}_2 ) is considered which represents the obstruction in order the fiber bundle be a tensorial product. Several properties and a splitting principle are proved for bundles [(E)\tilde]n\mathbbQ\tilde E_n^\mathbb{Q} . On this ground and founding on a convenient bundle BE → X associated to jaz (that we call Bonan's bundle and for which ɛ( [(E)\tilde]n\mathbbQ\tilde E_n^\mathbb{Q} =ɛ(BE)) relations are stated among Stiefel-Whitney classes of [(E)\tilde]n\mathbbQ\tilde E_n^\mathbb{Q} , BE and the class ɛ. - HYPERK¨AHLERHYPERK¨ HYPERK¨AHLER AND QUATERNIONIC K ¨ AHLER GEOMETRY, in support of application to supplicate for the degree of D. Phil. A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kähler. A similar result is proved for 8-manifolds. HyperKähler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kähler manifold (indefinite if the scalar curvature is negative). This construction is compatible with the quaternionic Kähler and hyperKähler quotient constructions and allows quater-nionic Kähler geometry to be subsumed into the theory of hyperKähler manifolds. It is shown that the hyperKähler metrics that arise admit a certain type of SU (2)-action, possess functions which are Kähler potentials for each of the complex structures simultaneously and determine quaternionic Kähler structures via a variant of the moment map construction. Quaternionic Kähler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKähler metrics with circle actions on complex line bundles over Kähler-Einstein (complex) contact manifolds. Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKähler metrics defined by Kronheimer, are shown to give rise to quaternionic Kähler met-rics and various examples of these metrics are identified. It is shown that any quaternionic Kähler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds. The twistor space structure of the projectivised nilpotent orbits is studied.