1.7. Isothermic surfaces 23 In other words, isothermic surfaces are characterized by the relations ∂1∂2f span(∂1f, ∂2f) and (1.71) ∂1f, ∂2f = 0, |∂1f|2 = α1s2, |∂2f|2 = α2s2, with a conformal metric coefficient s : R2 R+ and with the functions αi depending on ui only (i = 1, 2). These conditions may be equivalently represented as (1.72) ∂1∂2f = (∂2 log s)∂1f + (∂1 log s)∂2f, ∂1f, ∂2f = 0. Comparison with (1.25) shows that isothermic surfaces are nothing but or- thogonal Koenigs nets. Theorem 1.29. (Christoffel dual of an isothermic surface) Let f : R2 RN be an isothermic surface. Then the RN -valued one-form df∗ defined by (1.73) ∂1f∗ = α1 ∂1f |∂1f|2 = ∂1f s2 , ∂2f∗ = −α2 ∂2f |∂2f|2 = ∂2f s2 is closed. The surface f∗ : R2 RN , defined (up to a translation) by the integration of this one-form, is isothermic, with (1.74) ∂1f∗, ∂2f∗ = 0, |∂1f∗|2 = α1s−2, |∂2f∗|2 = α2s−2. The surface f∗ is called the Christoffel dual of the surface f. Conversely, if for a curvature line parametrized surface f∗ : R2 RN there exist functions αi = αi(ui) such that the one-form df∗ defined by (1.75) ∂1f∗ = α1 ∂1f |∂1f|2 , ∂2f∗ = −α2 ∂2f |∂2f|2 is closed, then the surface is isothermic, with the metric s defined by eq. (1.71). Another important class of transformations of isothermic surfaces is built by the Darboux transformations. Definition 1.30. (Darboux transformation) A Ribaucour transform f + : R2 RN of a given isothermic surface f : R2 RN is called a Darboux transform if its first fundamental form is conformal, upon the same reparametrization of independent variables ui ϕi(ui) (i = 1, 2) which makes the first fundamental form of f conformal, i.e., if |∂1f + |2/|∂2f + |2 = α1(u1)/α2(u2) holds at every point u R2 with the same functions αi as for f. Introduce the corresponding function s+ : R2 R+ for the surface f + , and denote r = s+/s : R2 R+. Thus, |∂if + |2/|∂if|2 = (s+/s)2 = r2
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