Connections and Curvature

  • Michael E. Taylor
Part of the Applied Mathematical Sciences book series (AMS, volume 116)


In this appendix we present results in differential geometry that serve as a useful background for material in the main body of the book. Material in §1 on connections is somewhat parallel to the study of the natural connection on a Riemannian manifold made in §11 of Chapter 1, but here we also study the curvature of a connection. Material in §2 on second covariant derivatives is connected with material in Chapter 2 on the Laplace operator. Ideas developed in §§3 and 4, on the curvature of Riemannian manifolds and submanifolds, make contact with such material as the existence of complex structures on two-dimensional Riemannian manifolds, established in Chapter 5, and the uniformization theorem for compact Riemann surfaces and other problems involving nonlinear, elliptic PDE, arising from studies of curvature, treated in Chapter 14. Section 5 on the Gauss-Bonnet theorem is useful both for estimates related to the proof of the unif ormization theorem and for applications to the Riemann-Roch theorem in Chapter 10. Furthermore, it serves as a transition to more advanced material presented in §§6–8.


Vector Field Riemannian Manifold Vector Bundle Covariant Derivative Fundamental Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AbM]
    R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin/Cummings, Reading, Mass., 1978.zbMATHGoogle Scholar
  2. [A1]
    C. Allendoerfer, The Euler number of a Riemannian manifold, Amer. J. Math. 62(1940), 243–248.MathSciNetCrossRefGoogle Scholar
  3. [AW]
    C. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannian polyhe-dra, Trans. AMS 53(1943), 101–129.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Ar]
    V. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.zbMATHGoogle Scholar
  5. [BGM]
    M. Berger, P. Gauduchon, and E. Mazet, Le Spectre d’une Variété Riemannienne, LNM no. 194, Springer-Verlag, New York, 1971.zbMATHGoogle Scholar
  6. [Bes]
    A. Besse, Einstein Manifolds, Springer-Verlag, New York, 1987.zbMATHGoogle Scholar
  7. [Car]
    M. doCarmo, Riemannian Geometry, Birkhauser, Boston, 1992.Google Scholar
  8. [Ch]
    J. Cheeger, Analytic torsion and the heat equation, Ann. Math. 109(1979), 259–322.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [CE]
    J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry, North Holland, Amsterdam, 1975.zbMATHGoogle Scholar
  10. [Cher]
    S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds., Ann. Math. 45(1944), 747–752.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [Chv]
    C. Chevalley, Theory of Lie Groups, Princeton Univ. Press, Princeton, N. J., 1946.zbMATHGoogle Scholar
  12. [EGH]
    T. Eguchi, P. Gilkey, and A. Hanson, Gravitation, Gauge Theories, and Differential Geometry, Physics Reports, Vol. 66, no. 6 (Dec. 1980).MathSciNetCrossRefGoogle Scholar
  13. [Fen]
    W. Fenchel, On total curvature of Riemannian manifolds, J. London Math. Soc. 15(1940), 15–22.MathSciNetCrossRefGoogle Scholar
  14. [FU]
    D. Freed and K. Uhlenbeck, Instantons and Four-Manifolds, Springer-Verlag, New York, 1984.zbMATHCrossRefGoogle Scholar
  15. [Gil]
    P. Gilkey, The Index Theorem and the Heat Equation, Publish or Perish, Boston, 1974.zbMATHGoogle Scholar
  16. [Gu]
    R. Gunning, Lectures on Riemann Surfaces, Princeton Univ. Press, Princeton, N. J., 1967.zbMATHGoogle Scholar
  17. [Hi]
    N. Hicks, Notes on Differential Geometry, Van Nostrand, New York, 1965.zbMATHGoogle Scholar
  18. [Jos]
    J. Lost, Nonlinear Methods in Riemannian and Kahlerian Geometry, Birkhauser, Boston, 1988.Google Scholar
  19. [Kaz]
    J. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Reg. Conf. Ser. in Math., no. 57, AMS, Providence, R. I., 1985.Google Scholar
  20. [KN]
    S. Kobayashi and N. Nomizu, Foundations of Differential Geometry, Interscience, New York, Vol. 1, 1963; Vol. 2, 1969.zbMATHGoogle Scholar
  21. [LM]
    H. B. Lawson and M. L. Michelson, Spin Geometry, Princeton Univ. Press, Princeton, N. J., 1989.zbMATHGoogle Scholar
  22. [Mi1]
    J. Milnor, Morse Theory, Princeton Univ. Press, Princeton, N. J., 1963.zbMATHGoogle Scholar
  23. [Mil2]
    J. Milnor, Topology from the Differentiable Viewpoint, Univ. Press of Virginia, Charlottesville, 1965.zbMATHGoogle Scholar
  24. [MiS]
    J. Milnor and J. Stasheff, Characteristic Classes, Princeton Univ. Press, Princeton, N. J., 1974.zbMATHGoogle Scholar
  25. [Pal]
    R. Palais, ed., Seminar on the Atiyah-Singer Index Theorem, Princeton Univ. Press, Princeton, N. J., 1963.Google Scholar
  26. [Poo]
    W. Poor, Differential Geometric Structures, McGraw-Hill, New York, 1981.zbMATHGoogle Scholar
  27. [Spi]
    M. Spivak, A Comprehensive Introduction to Differential Geometry, Vols. 1–5, Publish or Perish Press, Berkeley, Calif., 1979.Google Scholar
  28. [Stb]
    S. Sternberg, Lectures on Differential Geometry, Prentice Hall, Englewood Cliffs, N. J., 1964.zbMATHGoogle Scholar
  29. [Str]
    D. Struik, Lectures on Classical Differential Geometry, Addison-Wesley, Reading, Mass., 1964.Google Scholar
  30. [Wh]
    H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton, N. J., 1957.zbMATHGoogle Scholar
  31. [Wil]
    T. Wilmore, Total Curvature in Riemannian Geometry, Wiley, New York, 1982.Google Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Michael E. Taylor
    • 1
  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

Personalised recommendations