Difference between revisions of "Transport along a curve"
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| − | 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 
be a differential manifold, 
a vector bundle on . Let ![\gamma:[0,1] \to M](/w/images/math/9/f/4/9f4bf5bc55f62b0bbe89e8775d5fd2e2.png)
be a smooth curve in . Let 
denote a connection along 
defined by maps 
to the space of vector fields along ![\gamma<math>, denoted in symbols as:
<math>v \mapsto \phi_t(v) (t \in [0,1])](/w/images/math/1/1/f/11fa6e94c151a2b64487f174e7e7fafe.png)
such that for any vector 
:
and
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.
