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Abstract

Let π:Θ→M be a vector bundle with standard fiber ℝ d , dim M=n. Denote by Θ m the fiber at mM and by (m,ϑ)=ϑ m the points of this fiber. Consider a chart \(\mathcal{U}_{\alpha}\) on a manifold M with local coordinates (q 1,…,q n ) and a trivialization \(\mathcal{F}_{\alpha}\) of the bundle over that chart. Let e 1,…,e d be the standard basis in ℝ d . Since \(\mathcal{F}_{\alpha}(\pi^{-1}\mathcal{U}_{\alpha})=\mathcal{U}_{\alpha}\times \mathbb{R}^{d}\) , this basis generates a basis in every fiber Θ m , \(m\in \mathcal{U}_{\alpha}\) . We obtain a smooth field of bases that will also be denoted by e 1,…,e d . Thus every cross-section ϑ of the bundle Θ can be represented in terms of coordinates with respect to these bases in the form ϑ=ϑ i e i , i=1,…,d. In \(\mathcal{U}_{\alpha}\times \mathbb{R}^{d}\) the set of vectors \(\mathcal{U}_{\alpha}\times \{X_{0}\}\) for some X 0∈ℝ d corresponds to the vectors ϑ from Θ m , \(m\in \mathcal{U}_{\alpha}\) that have the same coordinates with respect to e 1,…,e d as X 0. Another trivialization of the bundle over \(\mathcal{U}_{\alpha}\) would generate another set of vectors equivalent to X 0 that is different from the former.

Keywords

Vector Bundle Tangent Space Covariant Derivative Tangent Bundle Principal Bundle 
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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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentVoronezh State UniversityVoronezhRussia

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