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| {{for|preliminary discussion|Cartan connection applications}}
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| In [[Riemannian geometry]], we can introduce a [[coordinate system]] over the [[Riemannian manifold]] (at least, over a [[Chart (topology)|chart]]), giving ''n'' coordinates
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| :<math>x_{i}\;\text{,}\qquad i = 1, \dots, n</math>
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| for an n-dimensional manifold. Locally, at least, this gives a basis for the 1-forms, dx<sup>i</sup> where d is the [[exterior derivative]]. The [[dual basis]] for the [[tangent space]] T is '''e'''<sub>i</sub>.
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| Now, let's choose an [[orthonormal basis]] for the [[fiber bundle|fibers]] of T. The rest is index manipulation.
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| ==Example==
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| Take a [[3-sphere]] with the [[radius]] ''R'' and give it [[polar coordinate]]s α, θ, φ.
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| :e('''e'''<sub>α</sub>)/R,
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| :e('''e'''<sub>θ</sub>)/R sin(α) and | |
| :e('''e'''<sub>φ</sub>)/R sin(α) sin(θ) | |
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| form an orthonormal basis of T.
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| Call these '''e'''<sub>1</sub>, '''e'''<sub>2</sub> and '''e'''<sub>3</sub>. Given the metric η, we can ignore the [[Covariance|covariant]] and [[contravariant]] distinction for T.
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| Then, the dreibein (triad),
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| :<math>e_1 = R\, d\alpha</math>
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| :<math>e_2 = R\, \sin{(\alpha)} d\theta</math>
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| :<math>e_3 = R\, \sin{(\alpha)} \sin{(\theta)} d\phi</math>. | |
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| So,
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| :<math>de_1=0</math>
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| :<math>de_2=R \cos{(\alpha)} d\alpha \wedge d\theta</math>
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| :<math>de_3=R (\cos{(\alpha)} \sin{(\theta)} d\alpha \wedge d\phi + \sin{(\alpha)} \cos{(\theta)} d\theta \wedge d\phi)</math>.
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| from the relation
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| :<math>d_\mathbf{A} e = de + A \wedge e = 0</math>,
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| we get
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| :<math>A_{12} = -\cos{(\alpha)} \, d\theta</math>
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| :<math>A_{13} = -\cos{(\alpha)} \, \sin{(\theta)} d\phi</math>
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| :<math>A_{23} = -\cos{(\theta)} \, d\phi</math>.
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| (d<sub>'''A'''</sub>η=0 tells us A is antisymmetric)
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| So, <math>\mathbf{F} = d\mathbf{A} + \mathbf{A} \wedge \mathbf{A}</math>,
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| :<math>F_{12}=\sin{(\alpha)} d\alpha\wedge d\theta</math>
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| :<math>F_{13}=\sin{(\alpha)} \sin{(\theta)} d\alpha\wedge d\phi</math>
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| :<math>F_{23}=\sin^2{(\alpha)} \sin{(\theta)} d\theta\wedge d\phi</math>
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| {{tensors}}
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| {{differential-geometry-stub}}
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| [[Category:Differential geometry]]
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| [[Category:Mathematical notation]]
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Production Manager (Forestry ) Ensley from Matachewan, has hobbies which include acting, new launch property singapore and rc model cars. Has enrolled in a global contiki tour. Is quite ecstatic particularly about traveling to Bagrati Cathedral and Gelati Monastery.
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