Transport along a curve: Difference between revisions

Revision as of 13:11, 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 $γ$ . The transport along $γ$ defined by $D / d t$ maps $E_{γ (0)}$ to the space of sections of $E$ along $γ$ , denoted in symbols as:

$v \mapsto ϕ_{t} (v) (t \in [0, 1])$

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

$ϕ_{0} (v) = v$

and

$\frac{D (ϕ_{t} (v)}{d t} = 0$

Intuitively, we define a rule for moving the fibre of $E$ , 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>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 T_{\gamma(0)}(M)</math>:
+such that for any vector <math>v \in E_{\gamma(0)})</math>:
 <math>\phi_0(v) = v</math>
@@ Line 13: / Line 13: @@
 <math>\frac{D (\phi_t(v)}{dt} = 0</math>
-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.
+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.