Abstract
This chapter begins with a standard elementary introduction to the theory of surfaces immersed in Euclidean space R 3, whose Riemannian metric is the standard dot product. Section 4.2 will be review for readers who have studied basic differential geometry of curves and surfaces in Euclidean space. Geometric intuition is used to construct Euclidean frames on a surface. Section 4.3 repeats the exposition, but this time following the frame reduction procedure outlined in Chapter 3 The classical existence and congruence theorems of Bonnet are stated and proved as consequences of the Cartan–Darboux Theorems. A section on tangent and curvature spheres provides needed background for Lie sphere geometry. The Gauss map helps tie together the formalism of Gauss and that of moving frames. We discuss special examples, such as surfaces of revolution, tubes about a space curve, inversions in a sphere, and parallel transforms of a given immersion. These constructions provide many valuable examples throughout the book. The latter two constructions introduce for the first time Möbius, respectively Lie sphere, transformations that are not Euclidean motions. The section on elasticae contains material needed in our introduction of the Willmore problems.
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Bonnet, P.O.: Mémoire sur la théorie des surfaces applicables sur une surface donnée, première partie. J. l’Ecole Polytech. 41, 209–230 (1866)
Bonnet, P.O.: Mémoire sur la théorie des surfaces applicables sur une surface donnée, deuxième partie. J. l’Ecole Polytech. 42, 1–151 (1867)
Bryant, R.L., Griffiths, P.: Reduction for constrained variational problems and \(\int \frac{1} {2}k^{2}\,ds\). Am. J. Math. 108(3), 525–570 (1986). doi:10.2307/2374654. http://www.dx.doi.org/10.2307/2374654
Calini, A., Ivey, T.: Bäcklund transformations and knots of constant torsion. J. Knot Theory Ramif. 7(6), 719–746 (1998). doi:10.1142/S0218216598000383. http://www.dx.doi.org/10.1142/S0218216598000383
Catalan, E.C.: Sur les surfaces réglés dont l’aire est un minimum. J. Math. Pure Appl. 7, 203–211 (1842)
Chern, S.S.: An elementary proof of the existence of isothermal parameters on a surface. Proc. Am. Math. Soc. 6, 771–782 (1955)
Chern, S.S.: Complex Manifolds Without Potential Theory (With an Appendix on the Geometry of Characteristic Classes). Universitext, 2nd edn. Springer, New York (1995).
Cieśliński, J.: The Darboux-Bianchi transformation for isothermic surfaces. Classical results versus the soliton approach. Differ. Geom. Appl. 7(1), 1–28 (1997). doi:10.1016/S0926-2245(97)00002-8. http://www.dx.doi.org.libproxy.wustl.edu/10.1016/S0926-2245(97)00002-8
Darboux, G.: Sur les surfaces isothermiques. C. R. Acad. Sci. Paris 128, 1299–1305 (1899)
Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.: Minimal Surfaces, I: Boundary Value Problems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 295. Springer, Berlin (1992)
Eells, J.: The surfaces of Delaunay. Math. Intell. 9(1), 53–57 (1987). doi:10.1007/BF03023575. http://www.dx.doi.org/10.1007/BF03023575
Gauss, C.F.: General Investigations of Curved Surfaces. Raven Press, Hewlett (1965)
Germain, S.: Recherches sur la Théorie des Surfaces Élastiques. Courcier, Paris (1821)
Griffiths, P.A.: Exterior Differential Systems and the Calculus of Variations. Progress in Mathematics, vol. 25. Birkhäuser, Boston (1983)
Hertrich-Jeromin, U., Pinkall, U.: Ein Beweis der Willmoreschen Vermutung für Kanaltori. J. Reine Angew. Math. 430, 21–34 (1992). doi:10.1515/crll.1992.430.21. http://www.dx.doi.org.libproxy.wustl.edu/10.1515/crll.1992.430.21
Langer, J., Singer, D.A.: The total squared curvature of closed curves. J. Differ. Geom. 20(1), 1–22 (1984)
Langer, J., Singer, D.A.: Lagrangian aspects of the Kirchhoff elastic rod. SIAM Rev. 38(4), 605–618 (1996)
Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218. Springer, New York (2003)
Levien, R.: The elastica: a mathematical history. http://levien.com/phd/elastica_hist.pdf (2008)
Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture. Ann. Math. (2) 179(2), 683–782 (2014). doi:10.4007/annals.2014.179.2.6. http://www.dx.doi.org/10.4007/annals.2014.179.2.6
Munkres, J.R.: Topology, 2nd edn. Prentice Hall, Upper Saddle River (2000)
Pinkall, U.: Hopf tori in S 3. Invent. Math. 81(2), 379–386 (1985). doi:10.1007/BF01389060. http://www.dx.doi.org.libproxy.wustl.edu/10.1007/BF01389060
Shepherd, M.D.: Line congruences as surfaces in the space of lines. Differ. Geom. Appl. 10(1), 1–26 (1999). doi:10.1016/S0926-2245(98)00025-4. http://www.dx.doi.org/10.1016/S0926-2245(98)00025-4
Shiohama, K., Takagi, R.: A characterization of a standard torus in E 3. J. Differ. Geom. 4, 477–485 (1970)
Thomsen, G.: Über konforme Geometrie I: Grundlagen der konformen Flächentheorie. Abh. Math. Sem. Hamburg 3, 31–56 (1924)
Truesdell, C.A.: The influence of elasticity on analysis: the classic heritage. Bull. Am. Math. Soc. (N.S.) 9(3), 293–310 (1983). doi:10.1090/S0273-0979-1983-15187-X. http://www.dx.doi.org/10.1090/S0273-0979-1983-15187-X
White, J.H.: A global invariant of conformal mappings in space. Proc. Am. Math. Soc. 38, 162–164 (1973)
Willmore, T.J.: Note on embedded surfaces. An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 11B, 493–496 (1965)
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Jensen, G.R., Musso, E., Nicolodi, L. (2016). Euclidean Geometry. In: Surfaces in Classical Geometries. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27076-0_4
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