Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems

Details Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems

2000-03-16

arxiv.org/abs/math/0003096v3

Integrable systems, Geometry and Topology (ed. C.-L. Terng) AMS/IP Studies in Advanced Math. vol. 36 (2006) 1-82

Mathematics

Differential Geometry

77 (A4) pages, 3 (small) PostScript figures. v3: Final final version: some typos corrected, references and exposition updated

Scientific paper

We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we identify Darboux transformations with the dressing action of "simple factors" in the sense of Terng and Uhlenbeck. In so doing, we advertise the use of Vahlen's Clifford algebra matrices as an efficient computational tool in conformal geometry.

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