Isothermic submanifolds of symmetric R-spaces

Abstract

Abstract. We extend the classical theory of isothermic surfaces in conformal 3-space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric R-spaces with essentially no loss of integrable structure. IntroductionBackground. A surface in R 3 is isothermic if, away from umbilics, it admits coordinates which are simultaneously conformal and curvature line or, more invariantly, if it admits a holomorphic quadratic differential q which commutes with the (trace-free) second fundamental form (then the coordinates are z = x + iy for which q = dz 2 ). Examples include surfaces of revolution, cones and cylinders; quadrics and constant mean curvature surfaces (where q is the Hopf differential).Starting with the work of Bour [9, §54], isothermic surfaces were the focus of intensive study by the geometers of the late 19th and early 20th centuries with contributions from Christoffel, Cayley, Darboux, Demoulin, Bianchi, Calapso, Tzitzéica and many others. There has been a recent revival of interest in the topic thanks to Cieśliński-Goldstein-Sym [26] who pointed out the links with soliton theory: indeed, isothermic surfaces constitute an integrable system with a particularly beautiful and intricate transformation theory, some aspects of which we list below:Conformal invariance. Since the trace-free second fundamental form is invariant (up to scale) under conformal diffeomorphisms of R 3 , such diffeomorphisms preserve the class of isothermic surfaces. Thus isothermic surfaces are more properly to be viewed as surfaces in the conformal 3-sphere.Deformations. At least locally, isothermic surfaces admit a 1-parameter deformation preserving the conformal structure and trace-free second fundamental form: this is the T -transformation of Bianchi [3] and Calapso [19]. In fact, isothermic surfaces are characterised by the existence of such a deformation [23] (see [47,17,14] for modern treatments).Darboux transformations. According to Darboux [27], given an isothermic surface, one may locally construct a 4-parameter family of new isothermic surfaces. Analytically, this is accomplished by solving an integrable system of linear differential equations; geometrically, the two surfaces envelop a conformal Ribaucour sphere congruence 1 . These transformations, the Darboux transformations, are analogous to the Bäcklund transformations of constant curvature surfaces (indeed, specialise to the latter in certain circumstances [40,43] Curved flats. Curved flats were introduced by and are an integrable system defined on submanifolds of a symmetric space G/H which, in non-degenerate cases, coincides with the G/H-system of Terng [54]. In particular, the space S 3 × S 3 \ ∆ of pairs of distinct points in S 3 is a (pseudo-Riemannian) symmetric space for the diagonal action of the conformal diffeomorphism group and curved flats in this space are the same as pairs of isothermic surfaces related by a Darboux transformation [16].Discrete theory. Bobenko-Pinkall [6] show that the combinatorics of ...

Special isothermic surfaces of type d

“…This action is transitive with parabolic stabilizers and each such stabilizer has abelian nilradical so that S n is a symmetric R space; cf. [4]. This viewpoint gives the conformal geometry of the sphere a projective linear flavour.…”

“…There is a gauge-theoretic formulation of the isothermic surface condition [4] that will be basic in all that follows. For this we begin by recalling the isomorphism…”

“…We have the following proposition. Proposition 1.11 [4,Lemma 3.13]. Let (Λ c , η c ) be a Christoffel transform of (Λ, η).…”

“…In recent times the subject has attracted new interest initiated by Cieśliński-Goldstein-Sym [9] who pointed out the relationship with integrable systems. Consequently, a highly developed modern theory of isothermic surfaces has emerged which inter alia extends the classical theory to arbitrary codimension and more exotic target spaces [3,4,5,14,16,17,20].…”

Covariant Description of Isothermic Surfaces

“…Most studies on isothermic surfaces refer to surfaces in a flat or conformally flat 3-dimensional space (see [16] and references therein for surfaces in symmetric spaces). Moreover, their definition is coordinate dependent.…”