Clifford analysis on spheres and hyperbolae. (English) Zbl 0909.30007

The consideration of Dirac operators over general manifolds is of great importance in different fields of applications, especially in geometry. Using Vahlen matrices the author succeeds to carry over a lot of results developed in in real Clifford analysis in \(\mathbb R^n\) to the sphere and the hyperbola. It seems to be one of the first papers which use Clifford analysis in the Minkowski space. I believe that it is a decisive step towards a general Clifford analysis in Krein spaces. Among other results the author proves the conformal covariance of the Dirac operator using the spin group associated to \(\mathbb R^{n,1}\).


30C35 General theory of conformal mappings
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
53C27 Spin and Spin\({}^c\) geometry
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[1] Ahlfors, Complex Variables 5 pp 215– (1986) · Zbl 0597.30062 · doi:10.1080/17476938608814142
[2] and , Elliptic Boundary Problems for Dirac Operators, Birkhauser, Boston, 1993. · doi:10.1007/978-1-4612-0337-7
[3] and , Clifford Analysis, Pitman, London, 1982.
[4] ’Clifford analysis for Dirac operators on manifolds with boundary’, to appear.
[5] Hurwitz pairs and applications of Möbius transformations, Thesis, Gent State University, Belgium, 1994.
[6] Dixon, Quart. J. Math. 35 pp 283– (1904)
[7] and , Quaternionic Analysis and Elliptic Boundary Value Problems, Birkhauser, Boston, 1990. · Zbl 0850.35001 · doi:10.1007/978-3-0348-7295-9
[8] Clifford Algebras and the Classical Groups, Cambridge University Press, Cambridge, 1995. · doi:10.1017/CBO9780511470912
[9] Qian, J. Operator Theory 35 pp 349– (1996)
[10] Ryan, Proc. Roy. Irish Acad. 85A pp 1– (1985)
[11] Ryan, Boletin de la Sociedard Matematica a Mexicana.
[12] Clifford analysis and Hardy 2-spaces on spheres and hyperbolae, Proceedings of the Symposium on Analytical and Numerical Methods in Quaternionic and Clifford Analysis, 153-188, Freiberg University Press, 1997.
[13] Sudbery, Math. Proc. Cambridge Phil. Soc. 85 pp 199– (1979) · Zbl 0399.30038 · doi:10.1017/S0305004100055638
[14] Vahlen, Math. Ann. 55 pp 585– (1902) · doi:10.1007/BF01450354
[15] ’Clifford analysis on the sphere’, to appear.
[16] Wada, Complex Variables 15 pp 125– (1990) · Zbl 0721.30035 · doi:10.1080/17476939008814442
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