## Abstract

Let *π*:Θ→*M* be a vector bundle with standard fiber ℝ^{ d }, dim *M*=*n*. Denote by Θ_{ m } the fiber at *m*∈*M* 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## Preview

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