# Difference between revisions of "Transport along a curve"

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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>. The transport along <math>\gamma<math> defined by <math>D/dt</math> maps <math>T_{\gamma(0)}(M)</math> to the space of [[vector field along a curve|vector field]]s 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>. The transport along <math>\gamma</math> defined by <math>D/dt</math> maps <math>T_{\gamma(0)}(M)</math> to the space of [[vector field along a curve|vector field]]s 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> |

## Revision as of 13:08, 1 September 2007

## Definition

Let be a differential manifold, a vector bundle on . Let be a smooth curve in . Let denote a connection along defined by maps to the space of vector fields along

such that for any vector :

and

Intuitively, we define a rule for *moving* the tangent space along the curve, in a manner that is parallel to itself with respect to the connection.