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.


Vector Bundle Tangent Space Covariant Derivative Tangent Bundle Principal Bundle 
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© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentVoronezh State UniversityVoronezhRussia

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