Transport along a curve

From Diffgeom
Revision as of 13:21, 1 September 2007 by Vipul (talk | contribs)
Jump to: navigation, search

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.