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Transport along a curve

388 bytes added, 18:02, 6 January 2012
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Let <math>M</math> be a [[differential manifold]], <math>E</math> a [[vector bundle]] on <math>M</math>. Let <math>\gamma:[0,1] \to M</math> be a [[smooth curve]] in <math>M</math>. Let <math>D/dt</math> denote a [[connection along a curve|connection along]] <math>\gamma</math> of <math>E</math>. The transport along <math>\gamma</math> defined by <math>D/dt</math> maps <math>E_{\gamma(0)}</math> to the space of [[section of a vector bundle along a curve|section]]s of <math>E</math> along <math>\gamma</math>, denoted in symbols as:
<math>v \mapsto \phi_t(v) (t \in [0,1])</math>
such that for any vector <math>v \in E_{\gamma(0)})</math>:
<math>\phi_0(v) = v</math>
<math>\frac{D (\phi_t(v))}{dt} = 0</math>
Intuitively, we define a rule for ''moving'' the fibre of <math>E</math>, in a manner that is parallel to itself with respect to the connection.
===Connection gives transport===
We saw above that a connection along a curve defines a rule of transport along that curve. Consider now a [[connection]] on the vector bundle. Then, for every curve, this gives rise to a connection along that curve, and hence a transport along that curve. Hence a connection gives a global rule for transporting [[frame]]s.
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