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The Penrose transform and Clifford analysis. (English) Zbl 0753.58033
Geometry and physics, Proc. Winter Sch., Srni/Czech. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 97-104 (1991).
[For the entire collection see Zbl 0742.00067.]
The Penrose transform is always based on a diagram of homogeneous spaces. Here the case corresponding to the orthogonal group $$SO(2n,C)$$ is studied by means of Clifford analysis [see F. Brackx, R. Delanghe and F. Sommen: Clifford analysis (1982; Zbl 0529.30001)], and is presented a simple approach using the Dolbeault realization of the corresponding cohomology groups and a simple calculus with differential forms (the Cauchy integral formula for solutions of the Laplace equation and the Leray residue for closed differential forms).
##### MSC:
 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 58A10 Differential forms in global analysis 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 58J05 Elliptic equations on manifolds, general theory 53C65 Integral geometry