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On Fueter-Hurwitz regular mappings. (English) Zbl 0864.30038

Corresponding to the style of the series this is a mostly self-contained presentation of some topics from quaternionic analysis – despite the fact that the author does not think a quaternionic analysis to exist. There are three main chapters, the first developing the necessary background from Fueter theory. The second chapter deals with quaternionic Kähler manifolds, e.g. the Lichnerowicz homotopy invariant is generalized to this case. A very interesting result is that harmonic mappings between Riemannian manifolds are directly and in a natural way combined with regular mappings in the Fueter sense.
At last the third chapter is dedicated to Hurwitz pairs and its connection to regular maps. A Hurwitz pair may be used to define a related Cauchy-Riemann operator, the kernel of which are the regular functions. The author calculates the polynomial solutions of the Fueter-Hurwitz equation and for general regular functions the Fourier and an integral representation. Also a supercomplex structure in the theory of pseudo-Euclidean Hurwitz pairs is defined and its properties are developed. The last pages show connections to Clifford algebras.
Added in 2006: The erratum corrects Theorem 2.6.2 on page 58. For the correct version and its proof the author refers to his forthcoming book in Springer Lecture Notes “The Fueter-Hurwitz operator and a Clifford-type structure”.
Reviewer: K.Habetha (Aachen)

MSC:

30G35 Functions of hypercomplex variables and generalized variables
53C20 Global Riemannian geometry, including pinching
15A63 Quadratic and bilinear forms, inner products
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