The problem of non-trivial isometries of surfaces preserving principal curvatures

Surfaces in E³ invariant under a one parameter group of isometries of E³

“…This isometry obviously preserves H = H (s) at the corresponding points. (See Roussos 1988b1999b, Soyuçok 1995. From these three facts we conclude that all helicoidal surfaces are Bonnet surfaces (see Roussos 1988b).…”

“…where t is parameter along the orbits of the group of the isometries (straight lines, circles, helices respectively) and s is parameter along the curves perpendicular to the orbits which are geodesics (see Baikoussis & Koufogiorgos 1997, 1998, Do Carmo & Dajczer 1982, Hitt & Roussos 1991, Eisenhart 1909, Soyuçok 1995 (19) gives…”

“…We observe that the constant C is the pitch of the helicoidal motion, h in Baikoussis & Koufogiorgos 1997, Do Carmo & Dajczer 1982, Soyuçok 1995 Changing ψ by a constant, for two constant values ψ 1 and ψ 2 of ψ we consider C 1 and C 2 constants satisfying…”

“…Also, there are three other isometric associate surfaces to a given isothermic helicoidal surface with the same mean curvature at the corresponding points of the isometries. (See Bobenko & Eitner 1998, Cartan 1942, Roussos 1988b1999a, Soyuçok 1995.…”

“…By definition, a surface M 2 in E 3 is called to be a Bonnet surface if it admits at least one non-trivial isometry from the surface to another surface or to itself that preserves the mean curvature (equivalent to, preseves each principal curvature). (See Bobenko & Eitner 1998, Cartan 1942, Roussos 1988b, 1999a, Soyuçok 1995, Voss 1993. It is a fact that if a surface admits two non-trivial and geometrically distinct isometries (that is, one is not the composition of the other followed by an isometry of the whole E 3 ) that preserve the mean curvature then it admits a whole one-parameter and differentiable family of such isometries and the surface is isothermic.…”