Transport along a curve: Difference between revisions

Revision as of 13:08, 1 September 2007

Definition

Let $M$ be a differential manifold, $E$ a vector bundle on $M$ . Let $γ : [0, 1] \to M$ be a smooth curve in $M$ . Let $D / d t$ denote a connection along $γ < m a t h > . T h e t r a n s p o r t a l o n g < m a t h > γ < m a t h > d e f i n e d b y < m a t h > D / d t$ maps $T_{γ (0)} (M)$ to the space of vector fields along $γ < m a t h >, d e n o t e d i n s y m b o l s a s : < m a t h > v \mapsto ϕ_{t} (v) (t \in [0, 1])$

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

$ϕ_{0} (v) = v$

and

$\frac{D (ϕ_{t} (v)}{d t} = 0$

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.

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 ==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>