Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kähler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-Kählerian and flat are found. It is proved that the quaternionic Kähler manifolds with the considered metric structure are Einstein for dimension at least 8. The c...

Let V = R be the pseudo-Euclidean vector space of signature (p, q), p ≥ 3 and W a module over the even Clifford algebra Cl0(V ). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V )-equivariant linear map Π : ∧2W → V . If the skew symmetric vector valued bilinear form Π is nondegenerate then (M,Q) is endowed with a canonical pseudo-Riemannian metric g such that (M,Q, g) is ...

Abstract. We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time transformations, we combine these one-dimensional spinors into bi-dimensional column vectors. From the transformation properties of the two-component ...

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CH, n ≥ 3. For the quaternionic hyperbolic spaces HH, n ≥ 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classificatio...

This paper studies the singularities of Cullen-regular functions of one quaternionic variable, as defined in [7]. The quaternionic Laurent series prove to be Cullen-regular. The singularities of Cullenregular functions are thus classified as removable, essential or poles. The quaternionic analogues of meromorphic complex functions, called semiregular functions, turn out to be quotients of Culle...

We classify the holomorphic structures of the tangent vertical bundle Θ of the twistor fibration of a quaternionic manifold (M, Q) of dimension 4n ≥ 8. Using a Penrose transform we show that, when (M, Q) is compact and admits a compatible quaternionic-Kähler metric of negative scalar curvature, Θ admits no global non-trivial holomorphic sections with respect to any of its holomorphic structures...

We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces Af with quaternionic multiplication attached to a normalized newform f without complex multiplication. We include an example of Af with quaternionic multiplication for which we find numerically a curve C whose Jacobian is Af up to numerical approximation, and we prove that it...

We present a new construction of translation invariant continuous valuations on convex compact subsets of a quaternionic space H n ≃ R4n. This construction is based on the theory of plurisubharmonic functions of quaternionic variables started by the author in [4] and [5] which is based in turn on the notion of non-commutative determinants. In this paper we also establish some new properties of ...

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g. [9, 10]. Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of H–planar curves coincides with the clas...

A manifold (M, I, J,K) is called hypercomplex if I, J,K are complex structures satisfying quaternionic relations. A quaternionic Hermitian hypercomplex manifold is called HKT (hyperkähler with torsion) if the (2,0)-form Ω associated with the corresponding Sp(n)-structure satisfies ∂Ω = 0. A Hermitian metric ω on a complex manifold is called balanced if d∗ω = 0. We show that balanced HKT metrics...