# Difference between revisions of "Transport along a curve"

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==Definition== | ==Definition== | ||

− | 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>. 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: | + | 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> | <math>v \mapsto \phi_t(v) (t \in [0,1])</math> | ||

− | such that for any vector <math>v \in E_{\gamma(0)} | + | such that for any vector <math>v \in E_{\gamma(0)}</math>: |

<math>\phi_0(v) = v</math> | <math>\phi_0(v) = v</math> | ||

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and | and | ||

− | <math>\frac{D (\phi_t(v)}{dt} = 0</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. | 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. | ||

+ | |||

+ | ==Facts== | ||

+ | |||

+ | ===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. |

## Latest revision as of 18:02, 6 January 2012

## Definition

Let be a differential manifold, a vector bundle on . Let be a smooth curve in . Let denote a connection along of . The transport along defined by maps to the space of sections of along , denoted in symbols as:

such that for any vector :

and

Intuitively, we define a rule for *moving* the fibre of , in a manner that is parallel to itself with respect to the connection.

## Facts

### 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 frames.