Transport along a curve: Difference between revisions

Latest revision as of 18:02, 6 January 2012

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 $\gamma$ of $E$ . The transport along $\gamma$ defined by $D/dt$ maps $E_{\gamma (0)}$ to the space of sections of $E$ along $\gamma$ , denoted in symbols as:

$v\mapsto \phi _{t}(v)(t\in [0,1])$

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

$\phi _{0}(v)=v$

and

${\frac {D(\phi _{t}(v))}{dt}}=0$

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

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.

@@ Line 1: / Line 1: @@
 ==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> 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>
-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 11: / Line 11: @@
 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 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.
+==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.