J Austral Math Soc Ser A 43 pp231--245, 1987.

Characterisation Theorems for Compact Hypercomplex Manifolds

S. Nag, J. A. Hillman and B. Datta

Abstract

We have defined and studied some pseudogroups of local diffeomorphims which generalise the complex analytic pseudogroups. A 4-dimensional (or 8-dimensional) manifold modelled on these 'Fueter 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-dimensional 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.

1980 AMS Subject Classification: 30G35, 32C99, 55R10, 58H05

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Authors

S. Nag

Mathematics/Statistics Division, Indian Statistical Institute, 203, B. T. Road, Calcutta 700035, India.

J. A. Hillman

School of Mathematics and Physics, Macquarie University, North Ryde, N.S.W. 2113, Australia.

B. Datta

Mathematics/Statistics Division, Indian Statistical Institute, 203, B. T. Road, Calcutta 700035, India.