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Characterisation theorems for compact hypercomplex manifolds

  • S. Nag (a1), J. A. Hillman (a2) and B. Datta (a3)

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 Further 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.

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References

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[1]Datta, B. and Nag, S., Some special classes of functions on Rn and their Jacobians Tech. Report No. 12/85, Indian Statistical Institute, 1985). To appear in Indian J. Pure Appl. Math. in revised form by Datta, B..
[2]Fueter, R., ‘Analytische Funktionen einer Quaternionen variablen’, Comment. Math. Helv. 4 (1932), 920.
[3]Nag, S., Laurent series over hypercomplex number systems (Tata Institute of Fundamental Research, Bombay, Research Report, 1983).
[4]Brackx, F., Delanghe, R. and Sommen, F., Clifford analysis (Pitman, Boston, Mass., 1982).
[5]Marchiafava, S., ‘Sulle varietà a struttura quaternionale generalizzata’, Rend. Mat. (6) 3 (1970), 529545.
[6]Salamon, S. M., ‘Quaternionic manifolds’, Symposia Mathematica 26 (1982), 139151.
[7]Sudbery, A., ‘Quaternionic analysis’, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199225.
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