Difference between revisions of "Transport along a curve"

From Diffgeom
Jump to: navigation, search
(Definition)
 
(5 intermediate revisions by the same user not shown)
Line 1: Line 1:
 
==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> 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>
 
<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>
 
<math>\phi_0(v) = v</math>
Line 11: Line 11:
 
and  
 
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.

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.