Difference between revisions of "Transport along a curve"

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(Definition)
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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>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 M be a differential manifold, E a vector bundle on M. Let \gamma:[0,1] \to M be a smooth curve in M. Let D/dt denote a connection along \gamma<math>. The transport along <math>\gamma<math> defined by <math>D/dt maps T_{\gamma(0)}(M) to the space of vector fields along \gamma<math>, denoted in symbols as:

<math>v \mapsto \phi_t(v) (t \in [0,1])

such that for any vector v \in T_{\gamma(0)}(M):

\phi_0(v) = v

and

\frac{D (\phi_t(v)}{dt} = 0

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.