Transport along a curve

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Revision as of 13:08, 1 September 2007 by Vipul (talk | contribs) (Definition)
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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<math>. The transport along <math>\gamma<math> defined by <math>D/dt maps T_{\gamma(0)}(M) to the space of vector fields along \gamma<math>, denoted in symbols as:

<math>v \mapsto \phi_t(v) (t \in [0,1])

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

\phi_0(v) = v

and

\frac{D (\phi_t(v)}{dt} = 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.