Beitr a-dieresis ge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 open parenthesis 2003 closing parenthesis comma No period .. 1 comma 1 hyphen 8 period Isothermic .. Surfaces .. and .. Hopf .. Cylinders Gabi Preissler Institute of Geometry comma .. Technical University of Dresden comma D hyphen 1 62 Dresden comma .. Germany e hyphen mail : preissler at math period tu hyphen dresden period de Abstract period .. Based .. on .. the .. work .. of Pinkall comma .. characterizations .. of spherical .. curves are given whose corresponding Hopf cylinders are isothermic surfaces in the three hyphen dimensional sphere period .. Comparing these characterizations with results of Langer and S inger about .. elastic spherical curves we determine all i sothermic Willmore Hopf tori period Keywords : .. isothermic surface comma Hopf cylinder comma Clifford torus comma Willmore surface 1 period Introduction In 1 985 comma U period Pinkall open square bracket 5 closing square bracket introduced Hopf cylinders in the three hyphen dimensional unit sphere S to the power of 3 subset R to the power of 4 which are inverse images of spherical curves in the two hyphen dimensional sphere S to the power of 2 subset R to the power of 3 by means of the Hopf map period .... Based on the work of J period Langer and D period A period .... S inger open square bracket 4 closing square bracket .... about elastic curves in S to the power of 2 comma Pinkall further determined all Willmore Hopf tori in S to the power of 3 period .... An example of a Willmore Hopf torus is the minimal Clifford torus in S to the power of 3 period .... This torus i s an isothermic surface : .... a suitable stereographic proj ection of it yields a torus of revolution in R to the power of 3 with a ratio of it s radii equal to 1 : square root of 2 sub period .. We can now ask whether there are any more i sothermic Willmore Hopf tori period .. Or more general : Which Hopf cylinders are isothermic ? In the next section we introduce i sothermic surfaces and in Section 3 we consider Hopf cylinders comma which can be described via quaternions open parenthesis see open square bracket 5 closing square bracket closing parenthesis period The answer to the above question i s given in Section 4 period .. There we characterize curves in the unit sphere S to the power of 2 which belong to isothermic Hopf cylinders period .... Namely comma the geodesic curvature of the spherical curves which correspond to isothermic Hopf cylinders is the tangent function 0 1 38 hyphen 482 1 slash 93 dollar 2 period 50 circlecopyrt-c 2003 Heldermann Verlag 2 .. G period Preissler : Isothermic Surfaces and Hopf Cylinders of a linear function of their arc length period .... Furthermore comma these curves are also characterized by a constant torsion in R to the power of 3 period Finally in Section 5 comma we apply the results to Willmore Hopf tori to determine all Willmore Hopf tori which are i sothermic period .. These are solely the minimal Clifford tori period 2 period Isothermic surfaces In the following comma .... all considered obj ects are assumed to be sufficiently differentiable comma .... e period g period .... in Section 4 we need differentiability class C to the power of 4 for the spherical curve period Definition 1 period .. A parametrization of a local surface is cal led is othermic if the components of the first fundamental form have the form g 1 1 = g 22 = lambda comma .. g 1 2 = 0 with a positive function lambda period .. An umbilicfree local surface is cal led an is othermic surface if there exists a parametrization of curvature lines on the surface which is is othermic period If the surface i s second order differentiable comma there always exists an isothermic parametrization open parenthesis cf period open square bracket 2 closing square bracket closing parenthesis period .. Further properties and examples of isothermic surfaces can be found e period g period in the books of B laschke open parenthesis open square bracket 1 closing square bracket comma p period 325 ff closing parenthesis and Eisenhart open parenthesis open square bracket 3 closing square bracket comma p period 1 7 f comma p period 2 26 ff closing parenthesis period .... A well hyphen known lemma giving an analytical criterion for an isothermic parametrization is Lemma 1 period .. On .. a .. surface .. there .. is .. an .. is othermic parametrization .. if and .. only .. if there .. are parameters open parenthesis u comma v closing parenthesis .. on the surface with partialdiff to the power of 2 divided by partialdiff u partialdiff v ln open parenthesis g 1 1 divided by g 22 to the power of closing parenthesis = 0 .. and .. g 1 2 = 0 period open parenthesis 1 closing parenthesis 3 period .. Hopf cylinders Here we cite some notations and results from open square bracket 5 closing square bracket in short period The unit sphere S to the power of 3 i s to be considered as a set of unit quaternions S to the power of 3 = open brace q in H bar bar q bar = 1 closing brace period S to the power of 2 can be described as the section of S to the power of 3 with a real three hyphen dimensional l inear subspace comma here we take S to the power of 2 = S to the power of 3 cap lin open brace 1 comma j comma k closing brace period From .. open square bracket 5 closing square bracket .. we know that the Hopf map pi : S to the power of 3 right arrow S to the power of 2 .. i s then given by pi open parenthesis q closing parenthesis = q-tilde q comma .. with pi open parenthesis e to the power of i phi q closing parenthesis = pi open parenthesis q closing parenthesis for all phi in R comma q in S to the power of 3 comma where q = q 0 plus q 1 to the power of i plus q 2 j plus q 3 to the power of k sub comma q-tilde = q 0 minus q 1 to the power of i plus q 2 j plus q 3 to the power of k sub comma q m in R comma .. m = 0 comma period period period comma 3 period Let p : .. open square bracket a comma b closing square bracket right arrow S to the power of 2 comma t arrowright-mapsto p open parenthesis t closing parenthesis comma be a regular spherical curve comma .. open square bracket a comma b closing square bracket subset equal R period .. We choose a curve y : .. open square bracket a comma b closing square bracket right arrow S to the power of 3 comma t mapsto-arrowright y open parenthesis t closing parenthesis comma with pi circ y = p period Definition 2 period .... The .... mapping x .... : .... open square bracket a comma b closing square bracket times S to the power of 1 right arrow S to the power of 3 .... with .... open parenthesis t comma phi closing parenthesis arrowright-mapsto e to the power of i phi y open parenthesis t closing parenthesis .... is .... cal led the .... Hopf cylinder of p in S to the power of 3 period .. If the spherical curve p is C to the power of 2 hyphen c los ed comma the Hopf cylinder of p is cal led Hopf torus of p period .. Especially comma .. if p is a circle comma .. we cal l the Hopf t orus of p a Clifford torus period G period Preissler : .. Isothermic Surfaces and Hopf Cylinders .. 3 Later comma for a geometrical interpretation we require Lemma s emi hyphen vertex from the equation to kappa sub g comma let k open parenthesis t closing parenthesis sub angle to the power of 2 period Let sub be the sub delta to the power of p sub open parenthesis to the power of be a spherical circle t closing parenthesis of tangent the delta sub open parenthesis t closing parenthesis = to the power of of curvature from cone to curve sub kappa sub g open parenthesis t closing parenthesis to the power of of p at t period Then for the from t ouching S to the power of 2 in the circ le k open parenthesis t closing parenthesis to with geodesic curvature holds period Figures 1 .... and 2 show a closed spherical curve p and a stereographic proj ection of it s Hopf torus in two different views period .... The parameter t can now be chosen as the arc length parameter Figure 1 period .. A closed spherical curve p with 6 periods Figure 2 period .. Stereographic image of the Hopf torus of p from Fig period .. 1 of y period .. For the arc length parameter s of p : .. open square bracket 0 comma l closing square bracket right arrow S to the power of 2 we get s = 2 t period According to open square bracket 5 closing square bracket comma the metric components and the components of the second fundamental form and the Weingarten map of a Hopf cylinder are then given by G = open parenthesis g ik closing parenthesis = Row 1 1 0 Row 2 0 1 . comma open parenthesis h sub ik closing parenthesis = open parenthesis h sub i to the power of k closing parenthesis = Row 1 2 kappa sub g 1 Row 2 1 0 . comma where kappa sub g i s the geodesic curvature of p period 4 .. G period Preissler : Isothermic Surfaces and Hopf Cylinders For the .. principal .. curvatures we .. obtain .. lambda sub 1 comma 2 = kappa sub g plusminux radicalbig-line of kappa sub g to the power of 2 plus 1 = kappa sub g plusminux kappa comma .. and .. we .. have v sub 1 comma 2 = x sub t plus open parenthesis minus kappa sub g plusminux kappa closing parenthesis x sub phi for the principal directions v sub 1 comma 2 and H = kappa sub g for the mean curvature H period So a Hopf cylinder has no umbilics comma and there exists a parametrization of curvature lines on it period 4 period Isothermic Hopf cylinders Now we want to apply Pinkall quoteright s calculus for Hopf cylinders to isothermic surfaces period Let the Hopf cylinder be parametrized by means of curvature l ine parameters open parenthesis u comma v closing parenthesis comma i period e period we have a curvature line parametrization x-tilde open parenthesis u comma v closing parenthesis = x open parenthesis t open parenthesis u comma v closing parenthesis comma phi open parenthesis u comma v closing parenthesis closing parenthesis with partial derivatives proportional to the principal curvature directions v sub 1 comma v sub 2 Line 1 x-tilde sub u = alpha open parenthesis u comma v closing parenthesis v sub 1 = partialdiff t divided by partialdiff u x sub t plus partialdiff phi divided by partialdiff u x sub phi Line 2 x-tilde sub v = beta open parenthesis u comma v closing parenthesis v sub 2 = partialdiff t divided by partialdiff v x sub t plus partialdiff phi divided by partialdiff v x sub phi comma where the proportionality factors alpha comma beta do not vanish period .. Here x sub t comma x sub phi denote the derivative of x with respect to t comma phi period .. After comparison of co efficients we obtain Equation: open parenthesis 2 closing parenthesis .. partialdiff t divided by partialdiff u = alpha comma partialdiff t divided by partialdiff v = beta and Equation: open parenthesis 3 closing parenthesis .. partialdiff phi divided by partialdiff u = alpha open parenthesis minus kappa sub g plus kappa closing parenthesis comma partialdiff phi divided by partialdiff v = minus beta open parenthesis kappa sub g plus kappa closing parenthesis period As we know that there locally exists a curvature l ine parametrization comma the following integra hyphen bility conditions must hold partialdiff divided by partialdiff v parenleftbigg partialdiff t divided by partialdiff u Row 1 partialdiff Row 2 = overbar partialdiff sub u . partialdiff t divided by partialdiff v parenrightbigg and partialdiff divided by partialdiff u parenleftbigg partialdiff phi divided by partialdiff v Row 1 partialdiff Row 2 = overbar partialdiff sub v . partialdiff phi divided by partialdiff u parenrightbigg period Inserting open parenthesis 2 closing parenthesis and open parenthesis 3 closing parenthesis comma the conditions are equivalent to .. open parenthesis kappa sub g to the power of prime : = d kappa sub g divided by dt comma kappa to the power of prime : = d kappa divided by dt closing parenthesis alpha sub v = beta sub u .. and beta sub u open parenthesis kappa sub g plus kappa closing parenthesis plus open parenthesis kappa sub g to the power of prime plus kappa to the power of prime closing parenthesis alpha beta plus alpha sub v open parenthesis minus kappa sub g plus kappa closing parenthesis plus open parenthesis minus kappa sub g to the power of prime plus kappa to the power of prime closing parenthesis alpha beta = 0 period The second equation can be transformed by the first one and we get Equation: open parenthesis 4 closing parenthesis .. alpha sub v = beta sub u and alpha sub v kappa plus kappa to the power of prime alpha beta = 0 period The matrix of metric components comma now in curvature line parametrization comma has the form Equation: open parenthesis 5 closing parenthesis .. G-tilde = open parenthesis tilde-g ik closing parenthesis = 2 kappa Row 1 alpha to the power of 2 open parenthesis kappa minus kappa sub g closing parenthesis 0 Row 2 0 beta to the power of 2 open parenthesis kappa plus kappa sub g closing parenthesis . where kappa sub g = kappa sub g open parenthesis t closing parenthesis comma kappa = kappa open parenthesis t closing parenthesis and t = t open parenthesis u comma v closing parenthesis period If we differentiate the second equation of open parenthesis 4 closing parenthesis partially with respect to v comma we obtain Equation: open parenthesis 6 closing parenthesis .. alpha sub vv kappa plus 2 alpha sub v beta kappa to the power of prime plus alpha beta sub v kappa to the power of prime plus alpha beta to the power of 2 kappa to the power of prime prime = 0 period Now we can show G period Preissler : .. Isothermic Surfaces and Hopf Cylinders .. 5 Proposition 1 period .... Exactly thos e Hopf cylinders .... are .... is othermic .... whose .... image .... under the Hopf map pi is a spherical curve p : open square bracket 0 comma l closing square bracket right arrow S to the power of 2 .. with geodesic curvature kappa sub g open parenthesis s closing parenthesis = tan open parenthesis c divided by 2 s plus d closing parenthesis where c comma d in R are suitab le constants and s is the arc length parameter of p period .. Especially comma .. the Clifford t ori open parenthesis c = 0 closing parenthesis are the only is othermic Hopf tori period Proof period .. If kappa to the power of prime vanishes everywhere comma then kappa = const and hence we have bar kappa sub g bar = square root of kappa to the power of 2 minus 1 = .. const period The curve p i s then a part of a circle and the assertion holds with c = 0 period So we can assume that we can find an interval with non hyphen vanishing kappa to the power of prime period .. We insert .. G-tilde in condition open parenthesis 1 closing parenthesis for an isothermic surface and we get 0 = partialdiff to the power of 2 divided by partialdiff u partialdiff v ln parenleftbigg g-tilde 1 1 divided by tilde-g 22 parenrightbigg = partialdiff to the power of 2 divided by partialdiff u partialdiff v open parenthesis 2 ln alpha minus 2 ln beta plus ln open parenthesis kappa minus kappa sub g closing parenthesis minus ln open parenthesis kappa plus kappa sub g closing parenthesis closing parenthesis open parenthesis with open parenthesis 5 closing parenthesis closing parenthesis Line 1 = partialdiff divided by partialdiff u parenleftbigg 2 alpha sub v divided by alpha minus 2 beta sub v divided by beta plus open parenthesis kappa minus kappa sub g closing parenthesis to the power of prime beta divided by kappa minus kappa sub g minus open parenthesis kappa plus kappa sub g closing parenthesis to the power of prime beta divided by kappa plus kappa sub g parenrightbigg Line 2 = partialdiff divided by partialdiff u parenleftbigg beta open parenthesis minus 2 kappa to the power of prime divided by kappa plus 2 open parenthesis kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis closing parenthesis minus 2 beta sub v divided by beta parenrightbigg using .. kappa to the power of 2 minus kappa sub g to the power of 2 = 1 .. and open parenthesis 4 closing parenthesis period .. Furthermore comma 0 = 2 parenleftbigg beta sub u open parenthesis minus kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis plus beta alpha open parenthesis minus kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis to the power of prime minus beta sub uv beta minus beta sub u beta sub v divided by beta to the power of 2 parenrightbigg = 2 parenleftbigg alpha sub v open parenthesis minus kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis minus alpha sub v kappa divided by kappa to the power of prime open parenthesis minus kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis to the power of prime minus alpha sub vv beta minus alpha sub v beta sub v divided by beta to the power of 2 parenrightbigg open parenthesis with open parenthesis 4 closing parenthesis closing parenthesis = 2 divided by beta to the power of 2 kappa parenleftbigg alpha sub v beta to the power of 2 kappa open parenthesis minus kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis minus alpha sub v beta to the power of 2 sub hline to the power of kappa kappa sub prime to the power of 2 open parenthesis minus kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis to the power of prime plus plus beta open parenthesis 2 alpha sub v beta kappa to the power of prime plus alpha beta sub v kappa to the power of prime plus alpha beta to the power of 2 kappa to the power of prime prime closing parenthesis plus alpha sub v beta sub v kappa parenrightbigg open parenthesis with open parenthesis 4 closing parenthesis and open parenthesis 6 closing parenthesis closing parenthesis = 2 divided by beta to the power of 2 kappa parenleftbigg alpha sub v beta to the power of 2 kappa open parenthesis kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis minus alpha sub v beta to the power of 2 sub hline to the power of kappa kappa sub prime to the power of 2 open parenthesis minus kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis to the power of prime plus plus open parenthesis minus alpha sub v beta to the power of 2 sub hline kappa sub prime to the power of kappa closing parenthesis kappa to the power of prime prime closing parenthesis open parenthesis with open parenthesis 4 closing parenthesis closing parenthesis Equation: open parenthesis 7 closing parenthesis .. = 2 alpha sub v divided by kappa kappa to the power of prime parenleftbigg kappa kappa to the power of prime open parenthesis kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis minus kappa to the power of 2 open parenthesis minus kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis to the power of prime minus kappa kappa to the power of prime prime parenrightbigg period From the equation .. kappa to the power of prime = kappa sub g kappa sub g to the power of prime divided by kappa open parenthesis Derivation of kappa = radicalbig-line of kappa sub g to the power of 2 plus 1 closing parenthesis as well as kappa to the power of prime prime = kappa sub g kappa sub g to the power of prime prime kappa to the power of 2 plus open parenthesis kappa sub g to the power of prime closing parenthesis to the power of 2 divided by kappa to the power of 3 6 .. G period Preissler : Isothermic Surfaces and Hopf Cylinders it follows Equation: open parenthesis 8 closing parenthesis .. open parenthesis kappa to the power of prime closing parenthesis to the power of 2 kappa = kappa sub g to the power of 2 open parenthesis kappa sub g to the power of prime closing parenthesis to the power of 2 divided by kappa and kappa to the power of prime kappa to the power of 2 = kappa sub g kappa sub g to the power of prime kappa and after differentiation we get Equation: open parenthesis 9 closing parenthesis .. open parenthesis minus kappa to the power of prime divided by kappa plus kappa sub g kappa to the power of prime minus kappa sub g to the power of prime kappa closing parenthesis to the power of prime = minus kappa to the power of prime prime divided by kappa plus open parenthesis kappa to the power of prime closing parenthesis to the power of 2 divided by kappa to the power of 2 plus kappa sub g kappa to the power of prime prime minus kappa sub g to the power of prime prime kappa period Inserting open parenthesis 9 closing parenthesis in open parenthesis 7 closing parenthesis and using open parenthesis 8 closing parenthesis we obtain Line 1 0 = alpha sub v parenleftbig open parenthesis kappa to the power of prime closing parenthesis to the power of 2 kappa kappa sub g minus kappa sub g to the power of prime kappa to the power of 2 kappa to the power of prime minus kappa sub g kappa to the power of prime prime kappa to the power of 2 plus kappa sub g to the power of prime prime kappa to the power of 3 parenrightbig Line 2 = alpha sub v parenleftbigg kappa sub g to the power of 3 open parenthesis kappa sub g to the power of prime closing parenthesis to the power of 2 divided by kappa minus kappa sub g open parenthesis kappa sub g to the power of prime closing parenthesis to the power of 2 kappa minus kappa sub g divided by kappa open parenthesis kappa sub g kappa sub g to the power of prime prime kappa to the power of 2 plus open parenthesis kappa sub g to the power of prime closing parenthesis to the power of 2 closing parenthesis plus kappa sub g to the power of prime prime kappa to the power of 3 parenrightbigg Line 3 = alpha sub v divided by kappa parenleftbig kappa sub g to the power of 3 open parenthesis kappa sub g to the power of prime closing parenthesis to the power of 2 minus kappa sub g open parenthesis kappa sub g to the power of prime closing parenthesis to the power of 2 kappa to the power of 2 minus kappa sub g to the power of 2 kappa sub g to the power of prime prime kappa to the power of 2 minus kappa sub g open parenthesis kappa sub g to the power of prime closing parenthesis to the power of 2 plus kappa sub g to the power of prime prime kappa to the power of 4 parenrightbig Line 4 = alpha sub v divided by kappa parenleftbig minus 2 kappa sub g open parenthesis kappa sub g to the power of prime closing parenthesis to the power of 2 plus kappa sub g to the power of prime prime open parenthesis 1 plus kappa sub g to the power of 2 closing parenthesis parenrightbig open parenthesis with several applications of kappa to the power of 2 = 1 plus kappa sub g to the power of 2 closing parenthesis = alpha sub v open parenthesis 1 plus kappa sub g to the power of 2 closing parenthesis to the power of 2 divided by kappa parenleftbigg kappa sub g to the power of prime divided by 1 plus kappa sub g to the power of 2 to the power of parenrightbigg prime period This equation is fulfilled iff alpha = alpha open parenthesis u closing parenthesis .... is only dependent on u or .... kappa prime sub g divided by 1 plus 2 kappa sub g = .... c holds with c = const period in R period .... In the first case comma we have kappa sub g = const period because of open parenthesis 4 closing parenthesis comma and p i s again a part of a circle period .. In the second case comma we integrate once more and get kappa sub g open parenthesis t closing parenthesis = tan open parenthesis ct plus d closing parenthesis comma where c comma d in R are suitable integration constants with bar ct plus d bar less pi divided by 2 and s = 2 t period square 4 period 1 period .. Characterizations of p corresponding to an isothermic Hopf cylinder Proposition 2 period .... A .... spherical curve p .... corresponding to .... an is othermic Hopf cylinder is .... char hyphen acterized by a constant torsion tau as a space curve in R to the power of 3 period .. For its curvature kappa we get kappa open parenthesis s closing parenthesis = 1 divided by cosine open parenthesis tau s plus d closing parenthesis period Proof period .... The assertion immediately follows with 2 tau = 2 d kappa sub g divided by ds divided by 2 kappa sub g plus 1 = kappa prime sub g divided by 2 kappa sub g plus 1 = c .... and .... kappa = radicalbig-line of kappa sub g to the power of 2 plus 1 after inserting the equations of Proposition 1 with bar tau s plus d bar less pi divided by 2 sub period square Proposition 3 period .... A spherical curve p corresponding to an is othermic Hopf cylinder is fu to the power of r-t her characterized by the fact that the s emi hyphen vertex angles .. of the .. cones touching S to the power of 2 .. in .. curvature circ les of p are linear functions of its arc length period Proof period .... A comparison of Proposition 1 with Lemma 2 yields delta open parenthesis s closing parenthesis = c divided by 2 s plus d = tau s plus d with c comma d in R suitably chosen period square Figure 3 shows the shape of p in S to the power of 2 for some values of c equal-negationslash 0 and d = 0 which is similar to a clothoid period .. Figure 4 presents a stereographic proj ection of the Hopf cylinder of p comma on the left hand side the whole surface comma on the right hand side a part of it period G period Preissler : .. Isothermic Surfaces and Hopf Cylinders .. 7 open parenthesis a closing parenthesis c = 2 open parenthesis b closing parenthesis c = 1 divided by 2 open parenthesis c closing parenthesis c = 1 divided by 4 Figure 3 period .... The spherical curve p from Proposition .... 1 with init ial conditions : .... p touches the equator for s = 0 and kappa sub g open parenthesis 0 closing parenthesis = 0 Figure 4 period .. Stereographic proj ection of the Hopf cylinder of the curve p from Fig period 3 comma .. open parenthesis b closing parenthesis 5 period Willmore Hopf tori Let M be a closed orientiable two hyphen dimensional manifold period .... Hence for simply C to the power of 2 hyphen closed spherical curves p we know from open square bracket 5 closing square bracket : f open parenthesis M closing parenthesis i s a Willmore Hopf torus iff p i s a crit ical point of oint kappa to the power of 2 open parenthesis s closing parenthesis ds period In 1987 comma such curves p called .. e lastic .. curves were found by Langer and S inger .. open square bracket 4 closing square bracket period .. They got the following result : Proposition 4 period .. There .. are .. infinitely .. many .. simply .. C to the power of infinity hyphen c los ed .. elastic .. curves p .. in .. S to the power of 2 .. with geodesic curvature kappa sub g open parenthesis s closing parenthesis = k sub 0 cn parenleftbigg k sub 0 s divided by 2 omega sub comma omega parenrightbigg comma where the maximal geodesic .... curvature k sub 0 .... is given by .... k sub 0 = square root of 2 omega divided by square root of 1 minus 2 omega to the power of 2 .... with a .... certain omega .... where omega to the power of 2 in open square bracket 0 comma 1 divided by 2 closing parenthesis .. and cn denotes the amplitude cosine .. open parenthesis Jacobi s el lipti c cosine closing parenthesis period Figures 1 and 2 show a closed elastic spherical curve for omega thickapprox 0 period 6894 open parenthesis k sub 0 thickapprox 4 period 3838 comma 6 periods closing parenthesis and a stereographic image of it s Willmore Hopf torus period A comparison of Propositions 1 and 4 yields for a C to the power of infinity hyphen closed p 8 .. G period Preissler : Isothermic Surfaces and Hopf Cylinders Corollary .... 1 period .... In .... the .... c lass .... of .... Willmore .... Hopf tori .... the .... minimal .... Clifford .... tori .... are .... the .... only is othermic surfaces period Proof period .. Both formulas for kappa sub g in Proposition 1 and Proposition 4 must coincide comma this is only possible for kappa sub g = 0 comma i period e period for p i s a great circle period .... The torus is therefore a minimal Clifford torus period square As the properties of being an i sothermic surface and of being a Willmore surface are M o-dieresis bius invariants comma we get Corollary 2 period .... Let f .... : S to the power of 3 right arrow S to the power of 3 .... be a .... conformal transformation comma .... M .... a Hopf torus .... and f open parenthesis M closing parenthesis an is othermic .. open parenthesis Willmore closing parenthesis surface period .. Then M is a open parenthesis minimal closing parenthesis .. Clifford t orus period Figures 1 to 4 were built with the computer algebra system Maple period References open square bracket 1 closing square bracket .. Blaschke W period : .. Vorlesungen u-dieresis ber Differentialgeometrie period Bd period I I I comma Springer 1929 period open square bracket 2 closing square bracket .. Chern comma S period semicolon Hartman comma P period semicolon Wintner comma A period : .. On Is othermic Coordinates period Comment period Math period Helv period 28 open parenthesis 1 954 closing parenthesis comma 301 endash 309 period .. Z b l 0 0 5 .. 6 period 40 2 0 6 open square bracket 3 closing square bracket .. Eisenhart comma L period P period : .. A .. Treatis e on the Differential Geometry of Curves and Surfaces period Dover Publications 1909 period .. Z .. b l .. 90 period 3 7 80 3 open square bracket 4 closing square bracket .... Langer comma .... J period semicolon .... S inger comma .... D period A period : .... Curve hyphen Straightening .... in Riemannian Manifolds period .... Ann period .... Global Anal period Geom period 5 open parenthesis 2 closing parenthesis .. open parenthesis 1987 closing parenthesis comma 133 endash 1 50 period .. Z b l 0 65 3 period 530 32 open square bracket 5 closing square bracket .. Pinkall comma U period : .. Hopf Tori in S to the power of 3 period Invent period Math period 81 .. open parenthesis 1 985 closing parenthesis comma 379 endash 386 period Z to the power of minus minus b l minus sub 0 58 to the power of minus 5 sub period to the power of minus 53 to the power of minus minus sub 0 5 to the power of minus 1 open square bracket 6 closing square bracket .. Pinkall comma U period semicolon Sterling I period : .. Willmore Surfaces period Math period Intelligencer 9 open parenthesis 2 closing parenthesis .. open parenthesis 1987 closing parenthesis comma 38 endash 43 period Z b l 0 6 16 period 5 3 0 49 open square bracket 7 closing square bracket .... Sterling comma .... I period : .... Willmore Surfaces and Computers period .... IMA Vol period .... Math period .... Appl period .... 5 1 .... open parenthesis 1993 closing parenthesis comma .... 13 1 endash 136 period .. Z b l .. 0 8 0 1 period 5 3 .. 0 0 5 Received May 10 comma 2001