Generalized quaternionic manifolds

We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein–Weyl space. In particular, on the product \(Z\) of any complex symplectic manifold \(M\) and the sphere, there exists a natural generalized complex structure, with respect to which \(Z\) is the twistor space of  \(M\) .     

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.     

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.     

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.     

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.     

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     

Permutation representations arising from simplicial complexes

Contents. Introduction. 1. The Lefschetz invariant of a G-complex. 2. Relative projeetivity and relative cohomology. 3. The Lefschetz invariant of a G-poset. 4. The lattice of all subgroups. 5. The lattice of subsets of a G-set. 6. The lattice of subspaces of a representation.     

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.

     

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     

Homology representations arising from the half cube

We construct a CW decomposition Cn of the n-dimensional half cube in a manner compatible with its structure as a polytope. For each 3?k?n, the complex Cn has a subcomplex Cn,k, which coincides with the clique complex of the half cube graph if k=4. The homology of Cn,k is concentrated in degree k−1 and furthermore, the (k−1)st Betti number of Cn,k is equal to the (k−2)nd Betti number of the complement of the k-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type Dn act naturally on the complexes Cn,k, and thus on the associated homology groups.     

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.      

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.     

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.     

Operators arising from representations of nilpotent Lie groups

The following two results are obtained for an irreducible multiplier representation T of a connected nilpotent Lie group. First, T_f is a Hilbert-Schmidt operator if f is square-integrable with compact support. Second, T_f is of trace class if f has derivatives with sufficiently many moments. An application is made of the latter result to show that T_f can be of trace class even when f is not continuous.     

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.