Basics of the Differential Geometry of Surfaces

Part of the Texts in Applied Mathematics book series (TAM, volume 38)


The purpose of this chapter is to introduce the reader to some elementary concepts of the differential geometry of surfaces. Our goal is rather modest: We simply want to introduce the concepts needed to understand the notion of Gaussian curvature, mean curvature, principal curvatures, and geodesic lines. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate differential geometry course offered in the fall of 1994. Most of the topics covered in this course have been included, except a presentation of the global Gauss–Bonnet–Hopf theorem, some material on special coordinate systems, and Hilbert’s theorem on surfaces of constant negative curvature.


Coordinate System Pattern Recognition Computer Image Differential Geometry Principal Curvature 
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  1. 1.
    Ralph Abraham and Jerrold E. Marsden. Foundations of Mechanics. Addison-Wesley, second edition, 1978.Google Scholar
  2. 2.
    Lars V. Ahlfors and Leo Sario. Riemann Surfaces. Princeton Math. Series, No. 2. Princeton University Press, 1960.MATHGoogle Scholar
  3. 3.
    Richard H. Bartels, John C. Beatty, and Brian A. Barsky. An Introduction to Splines for Use in Computer Graphics and Geometric Modelling. Morgan Kaufmann, first edition, 1987.Google Scholar
  4. 4.
    Marcel Berger and Bernard Gostiaux. G´eom´etrie diff´erentielle: vari´et´es, courbes et surfaces. Collection Math’ematiques. Puf, second edition, 1992. English edition: Differential geometry, manifolds, curves, and surfaces, GTM No. 115, Springer-Verlag.Google Scholar
  5. 5.
    William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, second edition, 1986.Google Scholar
  6. 6.
    Yvonne Choquet-Bruhat, C’ecile DeWitt-Morette, and Margaret Dillard-Bleick. Analysis, Manifolds, and Physics, Part I: Basics. North-Holland, first edition, 1982.Google Scholar
  7. 7.
    Gaston Darboux. Lec¸ons sur la th´eorie g´en´erale des surfaces, Troisieme Partie. Gauthier- Villars, first edition, 1894.Google Scholar
  8. 8.
    Gaston Darboux. Lec¸ons sur la th´eorie g´en´erale des surfaces, Quatrième Partie. Gauthier-Villars, first edition, 1896.Google Scholar
  9. 9.
    Gaston Darboux. Lec¸ons sur la th´eorie g´en´erale des surfaces, Première Partie. Gauthier-Villars, second edition, 1914.Google Scholar
  10. 10.
    Gaston Darboux. Lec¸ons sur la th´eorie g´en´erale des surfaces, Deuxième Partie. Gauthier-Villars, second edition, 1915.Google Scholar
  11. 11.
    Jean Dieudonn’e. Abr´eg´e d’Histoire des Math´ematiques, 1700–1900. Hermann, first edition, 1986.Google Scholar
  12. 12.
    Manfredo P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.Google Scholar
  13. 13.
    Manfredo P. do Carmo. Riemannian Geometry. Birkh‥auser, second edition, 1992.Google Scholar
  14. 14.
    Manfredo P. do Carmo. Differential Forms and Applications. Universitext. Springer-Verlag, first edition, 1994.Google Scholar
  15. 15.
    Gerald Farin. NURB Curves and Surfaces, from Projective Geometry to Practical Use. AK Peters, first edition, 1995.Google Scholar
  16. 16.
    Gerald Farin. Curves and Surfaces for CAGD. Academic Press, fourth edition, 1998.Google Scholar
  17. 17.
    William Fulton. Algebraic Topology, A First Course. GTM No. 153. Springer-Verlag, first edition, 1995.MATHGoogle Scholar
  18. 18.
    Jean H. Gallier. Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Morgan Kaufmann, first edition, 1999.Google Scholar
  19. 19.
    S. Gallot, D. Hulin, and J. Lafontaine. Riemannian Geometry. Universitext. Springer-Verlag, second edition, 1993.Google Scholar
  20. 20.
    R.V. Gamkrelidze (Ed.). Geometry I. Encyclopaedia of Mathematical Sciences, Vol. 28. Springer-Verlag, first edition, 1991.Google Scholar
  21. 21.
    Claude Godbillon. G´eom´etrie Diff´erentielle et M´ecanique Analytique. Collection M’ethodes. Hermann, first edition, 1969.Google Scholar
  22. 22.
    Andr’e Gramain. Topologie des Surfaces. Collection Sup. Puf, first edition, 1971.Google Scholar
  23. 23.
    A. Gray. Modern Differential Geometry of Curves and Surfaces. CRC Press, second edition, 1997.Google Scholar
  24. 24.
    Victor Guillemin and Alan Pollack. Differential Topology. Prentice-Hall, first edition, 1974.Google Scholar
  25. 25.
    D. Hilbert and S. Cohn-Vossen. Geometry and the Imagination. Chelsea Publishing Co., 1952.Google Scholar
  26. 26.
    Heinz Hopf. Differential Geometry in the Large. LNCS, Vol. 1000. Springer-Verlag, second edition, 1989.Google Scholar
  27. 27.
    J. Hoschek and D. Lasser. Computer-Aided Geometric Design. AK Peters, first edition, 1993.Google Scholar
  28. 28.
    Erwin Kreyszig. Differential Geometry. Dover, first edition, 1991.Google Scholar
  29. 29.
    Jacques Lafontaine. Introduction aux Vari´et´es Diff´erentielles. PUG, first edition, 1996.Google Scholar
  30. 30.
    Serge Lang. Differential and Riemannian Manifolds. GTM No. 160. Springer-Verlag, third edition, 1995.Google Scholar
  31. 31.
    Daniel Lehmann and Carlos Sacr’e. G´eom´etrie et Topologie des Surfaces. Puf, first edition, 1982.Google Scholar
  32. 32.
    Charles Loop. A G1 triangular spline surface of arbitrary topological type. Computer-Aided Geometric Design, 11:303–330, 1994.MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Paul Malliavin. G´eom´etrie Diff´erentielle Intrinsèque. Enseignement des Sciences, No. 14. Hermann, first edition, 1972.Google Scholar
  34. 34.
    William S. Massey. Algebraic Topology: An Introduction. GTM No. 56. Springer-Verlag, second edition, 1987.Google Scholar
  35. 35.
    William S. Massey. A Basic Course in Algebraic Topology. GTM No. 127. Springer-Verlag, first edition, 1991.Google Scholar
  36. 36.
    JohnW. Milnor. Topology from the Differentiable Viewpoint. The University Press of Virginia, second edition, 1969.Google Scholar
  37. 37.
    John W. Milnor. Morse Theory. Annals of Math. Series, No. 51. Princeton University Press, third edition, 1969.Google Scholar
  38. 38.
    Henry P. Moreton. Minimum curvature variation curves, networks, and surfaces for fair free-form shape design. PhD thesis, University of California, Berkeley, 1993.Google Scholar
  39. 39.
    Les Piegl and Wayne Tiller. The NURBS Book. Monograph in Visual Communications. Springer-Verlag, first edition, 1995.MATHGoogle Scholar
  40. 40.
    RichardW. Sharpe. Differential Geometry. Cartan’s Generalization of Klein’s Erlangen Program. GTM No. 166. Springer-Verlag, first edition, 1997.Google Scholar
  41. 41.
    S. Sternberg. Lectures On Differential Geometry. AMS Chelsea, second edition, 1983.Google Scholar
  42. 42.
    J.J. Stoker. Differential Geometry. Wiley Classics. Wiley-Interscience, first edition, 1989.Google Scholar
  43. 43.
    Frank Warner. Foundations of Differentiable Manifolds and Lie Groups. GTM No. 94. Springer-Verlag, first edition, 1983.Google Scholar
  44. 44.
    WilliamWelch. Serious Putty: Topological Design for Variational Curves and Surfaces. PhD thesis, Carnegie Mellon University, Pittsburgh, Pa., 1995.Google Scholar

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Authors and Affiliations

  1. 1.Department of Computer and Information ScienceUniversity of PennsylvaniaPhiladelphiaUSA

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