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De Rham Cohomology and Harmonic Differential Forms

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Riemannian Geometry and Geometric Analysis

Part of the book series: Universitext ((UTX))

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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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© 2002 Springer-Verlag Berlin Heidelberg

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Jost, J. (2002). De Rham Cohomology and Harmonic Differential Forms. In: Riemannian Geometry and Geometric Analysis. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04672-2_2

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  • DOI: https://doi.org/10.1007/978-3-662-04672-2_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42627-1

  • Online ISBN: 978-3-662-04672-2

  • eBook Packages: Springer Book Archive

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