Transport along a curve
From Diffgeom
Definition
Let be a differential manifold,
a vector bundle on
. Let
be a smooth curve in
. Let
denote a connection along
of
. The transport along
defined by
maps
to the space of sections of
along
, denoted in symbols as:
such that for any vector :
and
Intuitively, we define a rule for moving the fibre of , 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.