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De Rham Cohomology and Harmonic Differential Forms

  • Jürgen Jost
Chapter
  • 627 Downloads
Part of the Universitext book series (UTX)

Abstract

We need some preparations from linear algebra. Let V be a real vector space with a scalar product <·, ·>, and let Λ p V be the p-fold exterior product of V. We then obtain a scalar product on Λ p V by
$$\langle {\upsilon _1} \wedge ... \wedge {\upsilon _p},{\omega _1} \wedge ... \wedge {\omega _p}\rangle = \det \left( {\langle {\upsilon _i},{\omega _j}\rangle } \right)$$
(2.1.1)
and bilinear extension to Λ P (V). If e 1, ..., e d is an orthonormal basis of V,
$${e_{{i_1}}} \wedge ... \wedge {e_{{i_p}}}with1{i_1} < {i_2} < ... < {i_p}d$$
(2.1.2)
constitute an orthonormal basis of Λ p V.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Jürgen Jost
    • 1
  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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