Supplement on Curved Flats in the Space of Point Pairs and Isothermic Surfaces: A Quaternionic Calculus

A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. It introduces a rich algebraic structure and allows the use of global frames while, at the same time, incorporates the classical ``geometric'' model of Möbius geometry providing geometric clarity. This way, it provides the foundation for the development of new techniques in Möbius differential geometry.

A field where the quaternionic calculus already proved particularly useful is the geometry of transformations of isothermic surfaces: in the second half of the paper, the relation of Darboux and Christoffel pairs of isothermic surfaces and curved flats in the symmetric space of point pairs is discussed and some applications are sketched. In particular, a new viewpoint on relations between surfaces of constant mean curvature in certain spaces of constant curvature, and on Bryant's Weierstrass type representation for surfaces of constant mean curvature 1 in hyperbolic 3-space is presented.

1991 Mathematics Subject Classification: (Primary) 53A10, (Secondary)53A50, 53C42.

Keywords: Isothermic surface, Darboux transformation, Christoffel transformation, Goursat transformation, Curved flat, Constant mean curvature, Weierstrass representation.

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