quaternion-Kahler Research Articles

On Nearly-Kähler Geometry

We consider complete nearly-Kahler manifolds with a canonicalHermitian connection. We prove some metric properties of strict nearly-Kahler manifolds and give a sufficient condition for the reducibility of the canonical Hermitian connection. A holonomic condition for a nearly-Kahler manifold to be a twistor space over a quaternionic-Kahler manifold is given. This enables us to give classification results in 10-dimensions.

Hermitian and Kahler submanifolds of a quaternionic Kahler manifold

Hermitian and Kahler submanifolds of a quaternionic Kahler manifold

Hypermultiplets, hyperkahler cones and quaternion-Kahler geometry

We study hyperkahler cones and their corresponding quaternion-Kahler spaces. We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets. These manifolds characterize the geometry of the hypermultiplet sector of perturbative moduli spaces of type-II strings compactified on a Calabi-Yau manifold. As an example of our construction, we study the universal hypermultiplet in detail, and give three inequivalent tensor multiplet descriptions. We also comment on the construction of quaternion-Kahler manifolds that may describe instanton corrections to the moduli space.

Fano contact manifolds and nilpotent orbits

A contact structure on a complex manifold M is a corank 1 subbundle F of TM such that the bilinear form on F with values in the quotient line bundle L = TM/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.¶If ${\frak g}$ is a simple Lie algebra, the unique closed orbit in ${\bold P}({\frak g})$ (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.¶In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M $--\rightarrow$ P(H 0(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant.

Curvature properties of twistor spaces of quaternionic K�hler manifolds

We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kahler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kahler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kahler.

Spin c Manifolds and Complex Contact Structures

In this paper we extend our notion of projectable spinors ([9], Ch.1) to the framework of Spinc manifolds and deduce the basic formulas relating spinors on the base and the total space of Riemannian submersions with totally geodesic one-dimensional fibres. Some geometric applications concerning positive Kahler–Einstein complex contact manifolds (e.g. their characterisation as twistor spaces over positive quaternionic Kahler manifolds) are also given.

Compatible Almost Complex Structures on Quaternion Kähler Manifolds

Let (M4n,g,Q) be a quaternion Kahler manifold with reduced scalar curvature ν = K/4n(n + 2). Suppose J is an almost complex structure which is compatible with the quaternionic structure Q and let θ = − δ F ∘ J be the Lee form of J. We prove the following local results: (1) if J is conformally symplectic, then it is parallel and ν = 0; (2) if J is cosymplectic, then ν ≤ 0 with equality if and only if J is parallel; (3) if J is integrable, then dθ is Q-Hermitian and harmonic; and (4) any closed self-dual 2-form ω = f(g ∘ J) ∈ Λ2+ = g ∘ Q ⊂ Λ2 is parallel. In Section 5, extending previous results of Salamon [24], we describe a correspondence among conformally balanced J, Killing vector fields X and self-dual 2-forms μ satisfying the twistor equation.

Quaternionic Killing Spinors

In [17] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kahler manifolds. In the present article we study the limiting case, i.e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kahler manifolds with smallest possible eigenvalue are the quaternionic projective spaces.

The Geometry of Positive Locally Quaternion Kähler Manifolds

A classification of locally quaternion Kahler manifolds M4n with positive scalar curvature is obtained as a consequence of J. Wolf's work on space forms of irreducible symmetric spaces. We determine the Betti numbers of such manifolds M4n as well as of the “projective” 3-Sasakian manifolds fibering over them. We study the geometry of the quaternion Kahler and locally quaternion Kahler submanifolds for each M4n, which is particularly significant for 4n = 16 due to its relation with four quaternionic structures on the Grassmannian \(\overline {{\text{Gr}}} _4 \) (R8).

The First Eigenvalue of the Dirac Operator on Quaternionic Kähler Manifolds

In a previous paper we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kaehler 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 and show that a nontrivial solution defines a parallel spinor on the associated hyperkaehler manifold.

Harmonic manifolds with some specific volume densities

We show that a noncompact, complete, simply connected harmonic manifold (Md, g) with volume densityθm(r)=sinhd-1r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kahler harmonic manifold (M2d, g) with volume densityθm(r)=sinh2d-1r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kahler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented.

La première valeur propre de l'opérateur de Dirac sur les variétés kählériennes compactes

K.D. Kirchberg [Ki1] gave a lower bound for the first eigenvalue of the Dirac operator on a spin compact Kahler manifoldM of odd complex dimension with positive scalar curvature. We prove that manifolds of real dimension 8l+6 satisfying the limiting case are twistor space (cf. [Sa]) of quaternionic Kahler manifold with positive scalar curvature and that the only manifold of real dimension 8l+2 satisfying the limiting case is the complex projective spaceCP4l+1.

Quaternionic Kaehler manifolds and a curvature characterization of two-point homogeneous spaces

Quaternionic Kaehler manifolds and a curvature characterization of two-point homogeneous spaces

HyperK�hler and quaternionic K�hler geometry

 A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kahler. A similar result is proved for 8-manifolds. HyperKahler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kahler manifold (indefinite if the scalar curvature is negative). This construction is compatible with the quaternionic Kahler and hyperKahier quotient constructions and allows quaternionic Kahler geometry to be subsumed into the theory of hyperKahler manifolds. It is shown that the hyperKahler metrics that arise admit a certain type of SU (2)- action, possess functions which are Kahler potentials for each of the complex structures simultaneously and determine quaternionic Kahler structures via a variant of the moment map construction. Quaternionic Kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKahler metrics with circle actions on complex line bundles over Kahler-Einstein (complex) contact manifolds. Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKahler metrics defined by Kronheimer, are shown to give rise to quaternionic Kahler metrics and various examples of these metrics are identified. It is shown that any quaternionic Kahler 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.

Twistor theory and Blaschke manifolds

Using the twistor theory on quaternionic Kaehler manifolds and some recent results on Blaschke manifolds and compact manifolds whose holonomy group is Spin (7), we prove that a Blaschke manifold of nonnegative scalar curvature whose holonomy group is exceptional is isometric to a projective space.

Rigidity of nonnegatively curved compact quaternionic-Kähler manifolds

Soit M une variete de Kahler compacte quaternionique a courbure bisectionnelle quaternionique positive (non negative). Alors M est HP n avec la metrique standard (un espace symetrique quaternionique)

Yang-Mills fields on quaternionic spaces

A study is made of solutions of the Yang-Mills equations over a quaternionic Kahler manifold. The corresponding notion of self-duality is interpreted in terms of holomorphic geometry on a twistor space. Self-dual connections are constructed on various vector bundles over quaternionic projective spaces.

A Rigidity Theorem for Quaternionic-Kahler Manifolds

A Rigidity Theorem for Quaternionic-Kahler Manifolds

A generalization of the momentum mapping construction for quaternionic K�hler manifolds

We present a method of reduction of any quaternionic Kahler manifold with isometries to another quaternionic Kahler manifold in which the isometries are divided out. Our method is a generalization of the Marsden-Weinstein construction for symplectic manifolds to the non-symplectic geometry of the quaternionic Kahler case. We compare our results with the known construction for Kahler and hyperKahler manifolds. We also discuss the relevance of our results to the physics of supersymmetric non-linear σ-models and some applications of the method. In particular, we show that the Wolf spaces can be obtained as theU(1) andSU(2) quotients of quaternionic projective spaceHP(n). We also construct an interesting example of compact riemannianV-manifolds(orbifolds) whose metrics are quaternionic Kahler and not symmetric.

Connections for vector bundles over quaternionic Kähler manifolds

On definit des proprietes des connexions pour des fibres vectoriels sur des varietes de Kahler quaternioniques