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On Clifford manifolds of type Cl0, 3
by
Ilie Burdujan
Dept. of Mathematics, Univ. of Agriculture & Veterinary Medicine of Iasi - Romania
An almost hypercomplex structure of type CL0, 3 on an 8n-dimensional manifold M is a 6-tuple H = (J\alpha) (\alpha = [`1, 6]) of anticommuting almost complex structures J\alpha : TM --> TM satisfying the identities of cal CL0, 3, i.e. ( 1, J1, ..., J6, J7 = J1\circJ6) respect the multiplication table of CL0, 3. An almost Clifford structure of type CL0, 3 on M is a rank-7 subbundle Q subset End(TM) which is locally or globally spanned by almost hypercomplex structures H = (J\alpha). Consequently there exists a reduction of the structural group of the principal frame bundle of M to GL(n, CL0, 3)·(Sp1×Sp1) or GL(n, CL0, 3), respectively. Using the prolongation of the Lie algebra g corresponding to each such a Lie group, one can define the Spencer cohomology Hs, k(g). Then the corresponding obstructions to integrability are either only in H0, 2(g), H1, 2(g), H2, 2(g) or in H0, 2(g), H1, 2(g), respectively.
The set of all almost Clifford connections is determined.
The Kähler-Clifford manifolds are characterized by a non-zero harmonic $-form. Consequently if such a manifold is compact some informations about its Betti numbers are obtained.
Date received: March 14, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadz-85.