Integration on Manifolds

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

Abstract

After giving some definitions and results on orientability of smooth manifolds, the problems treated in the present chapter are concerned with orientation of smooth manifolds; especially the orientation of several manifolds introduced in the previous chapter, such as the cylindrical surface, the Möbius strip, and the real projective space ℝP2. Some attention is paid to integration on chains and integration on oriented manifolds, by applying Stokes’ and Green’s Theorems. Some calculations of de Rham cohomology are proposed, such as the cohomology groups of the circle and of an annular region in the plane. This cohomology is also used to prove that the torus T 2 and the sphere S 2 are not homeomorphic. The chapter ends with an application of Stokes’ Theorem to a certain structure on the complex projective space ℂP n .

Keywords

Projective Space Volume Element Compact Group Invariant Subspace Smooth Manifold 
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.
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References

  1. 1.
    Spivak, M.: Calculus on Manifolds. Benjamin, New York (1965) Google Scholar
  2. 2.
    Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics. Springer, Berlin (2010) Google Scholar

Further Reading

  1. 3.
    Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd revised edn. Academic Press, New York (2002) Google Scholar
  2. 4.
    Godbillon, C.: Éléments de Topologie Algébrique. Hermann, Paris (1971) Google Scholar
  3. 5.
    Hicks, N.J.: Notes on Differential Geometry. Van Nostrand Reinhold, London (1965) Google Scholar
  4. 6.
    Lee, J.M.: Manifolds and Differential Geometry. Graduate Studies in Mathematics. Am. Math. Soc., Providence (2009) Google Scholar
  5. 7.
    Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218. Springer, New York (2012) Google Scholar
  6. 8.
    Petersen, P.: Riemannian Geometry. Springer, New York (2010) Google Scholar
  7. 9.
    Spivak, M.: Differential Geometry, vols. 1–5, 3rd edn. Publish or Perish, Wilmington (1999) Google Scholar
  8. 10.
    Sternberg, S.: Lectures on Differential Geometry, 2nd edn. AMS Chelsea Publishing, Providence (1999) Google Scholar
  9. 11.
    Tu, L.W.: An Introduction to Manifolds. Universitext. Springer, Berlin (2008) 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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