Characterisation theorems for compact hypercomplex manifolds

Abstract

We have defined and studied some pseudogroups of local diffeomorphisms which generalise the complex analytic pseudogroups. A 4-dimensional (or 8-dimensional) manifold modelled on these 'Further pseudogroups' turns out to be a quaternionic (respectively octonionic) manifold. We characterise compact Fueter manifolds as being products of compact Riemann surfaces with appropriate dimensional spheres. It then transpires that a connected compact quaternionic (H) (respectively O) manifold X, minus a finite number of circles (its 'real set'), is the orientation double covering of the product Y × P², (respectively Y× P⁶), where Y is a connected surface equipped with a canonical conformal structure and Pⁿ is n-dimensonal real projective space. A corollary is that the only simply-connected compact manifolds which can allow H (respectively O) structure are S⁴ and S² × S² (respectively S⁸ and S²×S⁶). Previous authors, for example Marchiafava and Salamon, have studied very closely-related classes of manifolds by differential geometric methods. Our techniques in this paper are function theoretic and topological.