Lie applicable surfaces

Abstract

We give a detailed account of the gauge-theoretic approach to Lie applicable surfaces and the resulting transformation theory. In particular, we show that this approach coincides with the classical notion of Ω-and Ω 0surfaces of Demoulin.

Discrete $Ω$-nets and Guichard nets via discrete Koenigs nets

“…To summarise, we have the discrete analogue of the discussion in [38,Theorem 5.1] (c.f. [13, §2.5]):…”

“…Finally, one of us [38,Theorem 5.1] observed that Ω-surfaces may be similarly characterised in terms of two Combescure transforms x and ň, where the latter has principal curvatures 1 , 2 , for which…”

Surfaces in Laguerre geometry

“…Laguerre geometry is a classical sphere geometry that has its origins in the work of E. Laguerre in the mid 19th century and that had been extensively studied in the 1920s by Blaschke and Thomsen [6,7]. The study of surfaces in Laguerre geometry is currently still an active area of research [2,37,38,42,43,44,47,49,53,54,55] and several classical topics in Laguerre geometry, such as Laguerre minimal surfaces and Laguerre isothermic surfaces and their transformation theory, have recently received much attention in the theory of integrable systems [46,47,57,60,62], in discrete differential geometry, and in the applications to geometric computing and architectural geometry [8,9,10,58,59,61].…”

Darboux and Calapso transforms of meromorphically isothermic surfaces

“…Already in the nineteenth century geometers discovered methods to construct new surfaces from a given one while preserving some geometrical properties, such as the Combescure transformation, which preserves tangent planes up to translations, and the Ribaucour transformation, which preserves curvature lines and an enveloped sphere congruence, see [Bia22] and [Eis62], or [DT03] for a more modern treatment. While these transformations exist for arbitrary smooth surfaces in Euclidean space, others are defined only on subclasses of smooth surfaces, such as the Bäcklund and Lie transformations of pseudospherical surfaces (see [Eis60]) in Euclidean geometry or the Christoffel, Darboux and Calapso transformations of isothermic surfaces in Möbius geometry (see [Bur06,HJ03]) and Lie applicable surfaces in Lie sphere geometry (see [Pem16]). During the last three decades, many of these surfaces have been shown to constitute integrable systems by relating their transformations to a pencil of flat connections, as summarized in [Bur17], cf.…”

“…For example, the Weierstrass representation of a minimal surface can be interpreted as the Goursat transform of its Weierstrass data, as explained in [HJ03,HJH17], and Bryant's representation [Bry87] of CMC-1 surfaces in hyperbolic space can be related to the Darboux transformation of isothermic surfaces, see [HJMN01]. Recently, these and further representation formulas have been linked to the transformation theory of Ω-surfaces in [Pem16].…”