Generalised connected sums of quaternionic manifolds

Using deformations of singular twistor spaces, a generalisation of the connected sum construction appropriate for quaternionic manifolds is introduced. This is used to construct examples of quaternionic manifolds which have no quaternionic symmetries and leads to examples of quaternionic manifolds whose twistor spaces have arbitrary algebraic dimension.Partially supported by the National Science Foundation grant DMS-9296168.     

L-S category of quaternionic Stiefel manifolds

By calculating certain generalized cohomology theory, lower bounds for the L-S category of quaternionic Stiefel manifolds are given.     

A new class of quaternionic manifolds

Riemannian manifolds with structure group Sp(n)·Sp(1) are studied. Moreover for the Riemannian manifolds with structure group Sp(n) a new class is defined, and example of compact nilmanifolds in this class are constructed.     

A direct approach to quaternionic manifolds

The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on , in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.     

The Yamabe problem on quaternionic contact manifolds

By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(ℍ ⁿ ). Mathematics Subject Classification (2000) 53C17, 53D10, 35J70     

Compatible complex structures on almost quaternionic manifolds

On an almost quaternionic manifold we study the integrability of almost complex structures which are compatible with the almost quaternionic structure . If , we prove that the existence of two compatible complex structures forces to be quaternionic. If , that is is an oriented conformal 4-manifold, we prove a maximum principle for the angle function of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure on the twistor space of an almost quaternionic manifold and show that is a complex structure if and only if is quaternionic. This is a natural generalization of the Penrose twistor constructions.

     

New HKT manifolds arising from quaternionic representations

We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K?hler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in ${SL(n,\mathbb{H})}$ which is not a nilmanifold. We find in addition new compact strong HKT manifolds. We also show that every K?hler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional K?hler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8. Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics.     

Twistorial transformation of conformally einstein quaternionic manifolds

Department of Theoretical Problems, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 4, pp. 84–85, October–December, 1990.     

Einstein metrics and quaternionic Kähler manifolds

The research is partially suported by a grant from the Natural Sciences and Engineering Research Council of Canada     

Generalized planar curves and quaternionic geometry

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., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), 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 class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.     

Rigidity and vanishing theorems for almost quaternionic manifolds

We prove the rigidity under (compatible) circle actions of several twisted Dirac operators on almost quaternionic manifolds, and the vanishing of the indices of some of them as a consequence.      

Twistor theory for co-CR quaternionic manifolds and related structures

In a general and non-metrical framework, we introduce the class of co-CR quaternionic manifolds, which contains the class of quaternionic manifolds, whilst in dimension three it particularizes to give the Einstein-Weyl spaces. We show that these manifolds have a rich natural Twistor Theory and, along the way, we obtain a heaven space construction for quaternionic-Kähler manifolds.     

Generalized Trans-Sasakian manifolds

In this paper, we study the class of almost contact metric manifolds which are conformal to Trans-Sasakian manifolds, and we construct concrete examples from almost Hermitian manifolds using the product of manifolds. As a consequence, we obtain several properties for the three-dimensional case.     

Twistor theory for CR quaternionic manifolds and related structures

In a general and nonmetrical framework, we introduce the class of CR quaternionic manifolds containing the class of quaternionic manifolds, whilst in dimension three it particularizes to, essentially, give the conformal manifolds. We show that these manifolds have a rich natural Twistor Theory and, along the way, we obtain a heaven space construction for quaternionic manifolds.     

Generalized Whittaker vectors for holomorphic and quaternionic representations

The conjugacy class of parabolic subgroups with Heisenberg unipotent radical in a simple Lie groups over ³ not of type C_nC_{n} contains an element defined over  for each quaternionic real form. In this paper we study the Whittaker models for quaternionic discrete series of these real forms and prove results analogous and by analogous methods to the case of simple Lie groups over  that are the automorphism groups of tube type Hermitian symmetric domain and (so-called Bessel models) for holomorphic representations. In particular we calculate the decomposition of the space of Whittaker vectors under the action of the stabilizer of the corresponding character in a Levi factor of the Heisenberg parabolic subgroup.     

Generalized quaternionic involute-evolute curves in the Euclidean four-space E4

In this paper, a new type of quaternionic partner curves is defined as generalized quaternionic involute-evolute curves or $(0,2)$-quaternionic involute– $(1,3)$-quaternionic evolute curves in the four-dimensional Euclidean space. The relations between the Frenet frames and curvatures of the quaternionic involute-evolute curve couple are introduced. Moreover, the necessary and sufficient conditions for a quaternionic curve to have a generalized involute are obtained.