Tensor Fields and Differential Forms

  • Pedro M. Gadea
  • Jaime Muñoz Masqué
  • Ihor V. Mykytyuk
Part of the Problem Books in Mathematics book series (PBM)


After providing some definitions and results on tensor fields and differential forms, this chapter deals with some aspects of general vector bundles, including the ‘cocycle approach’; other topics are: Tensors and tensor fields, exterior forms, Lie derivative and the interior product; calculus of differential forms and distributions. Some examples related to manifolds studied in the previous chapter are also present, such as the infinite Möbius strip, considered as a vector bundle, and the tautological bundle over the real Grassmannian. Certain problems intend to make the reader familiar with computations of vector fields, differential forms, Lie derivative, the interior product, the exterior differential, and their relationships. Other group of problems tries to develop practical abilities in computing integral distributions and differential ideals.


Vector Field Vector Bundle Differential Form Kind Permission Differential Ideal 
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  1. 1.
    Gawedzki, K.: Fourier-Like Kernels in Geometric Quantization. Dissertationes Math., vol. 1284 (1976), 83 pp. Google Scholar
  2. 2.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vols. I, II. Wiley Classics Library. Wiley, New York (1996) Google Scholar
  3. 3.
    Lawson, H.B.: Foliations. Bull. Am. Math. Soc. 80(3), 369–418 (1974) zbMATHCrossRefGoogle Scholar
  4. 4.
    Reeb, G.: sur Certaines Propriétés Topologiques des Variétés Feuillétées. Actualités Sci. Indust., vol. 1183. Hermann, Paris (1952) zbMATHGoogle Scholar

Further Reading

  1. 5.
    Bishop, R.L., Crittenden, R.J.: Geometry of Manifolds. AMS Chelsea Publishing, Providence (2001) zbMATHGoogle Scholar
  2. 6.
    Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd revised edn. Academic Press, New York (2002) Google Scholar
  3. 7.
    Brickell, F., Clark, R.S.: Differentiable Manifolds. Van Nostrand Reinhold, London (1970) zbMATHGoogle Scholar
  4. 8.
    Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Springer, Berlin (2004) zbMATHCrossRefGoogle Scholar
  5. 9.
    Godbillon, C.: Géométrie Différentielle et Mécanique Analytique. Hermann, Paris (1969) Google Scholar
  6. 10.
    Hicks, N.J.: Notes on Differential Geometry. Van Nostrand Reinhold, London (1965) zbMATHGoogle Scholar
  7. 11.
    Lee, J.M.: Manifolds and Differential Geometry. Graduate Studies in Mathematics. Am. Math. Soc., Providence (2009) zbMATHGoogle Scholar
  8. 12.
    Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218. Springer, New York (2012) CrossRefGoogle Scholar
  9. 13.
    Lichnerowicz, A.: Global Theory of Connections and Holonomy Groups. Noordhoff, Leyden (1976) zbMATHCrossRefGoogle Scholar
  10. 14.
    Lichnerowicz, A.: Geometry of Groups of Transformations. Noordhoff, Leyden (1977) zbMATHGoogle Scholar
  11. 15.
    O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983) zbMATHGoogle Scholar
  12. 16.
    Spivak, M.: Differential Geometry, vols. 1–5, 3nd edn. Publish or Perish, Wilmington (1999) zbMATHGoogle Scholar
  13. 17.
    Sternberg, S.: Lectures on Differential Geometry, 2nd edn. AMS Chelsea Publishing, Providence (1999) Google Scholar
  14. 18.
    Tu, L.W.: An Introduction to Manifolds. Universitext. Springer, Berlin (2008) zbMATHGoogle Scholar
  15. 19.
    Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics. Springer, Berlin (2010) Google Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Pedro M. Gadea
    • 1
  • Jaime Muñoz Masqué
    • 2
  • Ihor V. Mykytyuk
    • 3
    • 4
  1. 1.Instituto de Física FundamentalCSICMadridSpain
  2. 2.Instituto de Seguridad de la InformaciónCSICMadridSpain
  3. 3.Institute of MathematicsPedagogical University of CracowCracowPoland
  4. 4.Institute of Applied Problems of Mechanics and MathematicsNASUL’vivUkraine

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