Abstract
Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and L-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new Bäcklund-type transformation for these surfaces.
Highlights
In [14,15,16], Demoulin defined a class of surfaces satisfying the equation √ V √E κ1,u + 2 U √G κ2,v = 0, (1) U G κ1 − κ2 vV E κ1 − κ2 u given in terms of curvature line coordinates (u, v), where U is a function of u, V is a function of v, ∈ {0, 1, i}, E and G denote the usual coefficients of the first fundamental form and κ1 and κ2 denote the principal curvatures
By using the hexaspherical coordinate model of Lie [27] it is shown in [31]. That these surfaces are the deformable surfaces of Lie sphere geometry
Special -surfaces of type 1 whose linear polynomial conserved quantity p satisfies ( p(t), p(t)) = 0 are the L-isothermic surfaces of any Laguerre geometry defined by p(0)
Summary
V E κ1 − κ2 u given in terms of curvature line coordinates (u, v), where U is a function of u, V is a function of v, ∈ {0, 1, i}, E and G denote the usual coefficients of the first fundamental form and κ1 and κ2 denote the principal curvatures. That these surfaces are the deformable surfaces of Lie sphere geometry This gives rise to a gauge theoretic approach for these surfaces which is developed in [13]. The definition of Lie applicable surfaces is equated to the existence of a certain 1-parameter family of flat connections This approach lends itself well to the. By considering polynomial conserved quantities of the arising 1-parameter family of flat connections, one can characterise familiar subclasses of surfaces in certain space forms. In [8,9] linear Weingarten surfaces in space forms are characterised as Lie applicable surfaces whose isothermic sphere congruences take values in certain sphere complexes. By using this approach we obtain a new Bäcklund-type transformation for linear Weingarten surfaces
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