Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Monogenic differential calculus

Author: F. Sommen
Journal: Trans. Amer. Math. Soc. 326 (1991), 613-632
MSC: Primary 30G35; Secondary 58A10
MathSciNet review: 1012510
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study differential forms satisfying a Dirac type equation and taking values in a Clifford algebra. For them we establish a Cauchy representation formula and we compute winding numbers for pairs of nonintersecting cycles in $ {\mathbb{R}^m}$ as residues of special differential forms. Next we prove that the cohomology spaces for the complex of monogenic differential forms split as direct sums of de Rham cohomology spaces. We also study duals of spaces of monogenic differential forms, leading to a general residue theory in Euclidean space. Our theory includes the one established in our paper [11] and is strongly related to certain differential forms introduced by Habetha in [4].

References [Enhancements On Off] (What's this?)

  • [1] J. W. Alexander, On the chains of a complex and their duals, Proc. Nat. Acad. Sci. U.S.A. 21 (1935), 509-511.
  • [2] F. Brackx, R. Delanghe, and F. Sommen, Clifford anaylsis, Research Notes in Math., no. 76, Pitman, London, 1982.
  • [3] R. Delanghe and F. Brackx, Duality in hypercomplex function theory, J. Funct. Anal. 37 (1978), 164-181. MR 578930 (81j:46029a)
  • [4] K. Habetha, Eine Definition des Kroneckerindexes im $ {\mathbb{R}^{m + 1}}$ mit Hilfe der Cliffordanalysis, Z. Anal. Anwendungen 5 (1986), 133-137. MR 837640 (87h:30098)
  • [5] D. Hestenes and G. Sobczyk, Clifford algebra to geometric calculus, Reidel, Dordrecht, 1984. MR 759340 (86g:15012)
  • [6] M. W. Hirsch, Differential topology, Graduate Texts in Math., vol. 33, Springer, New York, 1976. MR 0448362 (56:6669)
  • [7] W. Hodge, The theory and applications of harmonic integrals, Cambridge Univ. Press, 1959.
  • [8] L. S. Pontryagin, Foundations of combinatorial topology, Graylock Press, Rochester, 1952. MR 0049559 (14:194b)
  • [9] G. de Rham, Variétés différentiables. Formes, courants, formes harmoniques, Hermann, Paris, 1955.
  • [10] F. Sommen, An extension of the Radon transform to Clifford analysis, Complex Variables Theory Appl. 8 (1987), 249-266. MR 898067 (88j:30096)
  • [11] -, Monogenic differential forms and homology theory, Proc. Roy. Irish Acad. 84 (1984), 87-109. MR 790302 (87b:30074)
  • [12] F. Sommen and V. Souček, Hypercomplex differential forms applied to the de Rham and the Dolbeault complex, Sem. Geom. 1984, Univ. Bologna, 1985, pp. 177-192. MR 866157 (88a:58008)
  • [13] V. Souček, Quaternion valued differential forms in $ {\mathbb{R}^4}$, Suppl. Rend. Circ. Mat. Palermo (2) 33 (1984), 293-300.
  • [14] C. von Westenholz, Differential forms in mathematical physics, Stud. Math. Appl., vol. 3 North-Holland, Amsterdam, 1978. MR 0494187 (58:13108)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30G35, 58A10

Retrieve articles in all journals with MSC: 30G35, 58A10

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society