Editing Parallel transport (section)

==Parallel transport on a vector bundle==
Parallel transport of tangent vectors is a special case of a more general construction involving an arbitrary [[vector bundle]] <math>E</math>. Specifically, parallel transport of tangent vectors is the case where <math>E</math> is the [[tangent bundle]] <math>TM</math>.

Let ''M'' be a smooth manifold. Let ''E''&nbsp;&rarr;&nbsp;''M'' be a vector bundle with [[connection (vector bundle)|connection]] &nabla; and ''&gamma;'': ''I''&nbsp;&rarr;&nbsp;''M'' a [[curve|smooth curve]] parameterized by an open interval ''I''. A [[Section (fiber bundle)|section]] <math>X</math> of <math>E</math> along ''&gamma;'' is called '''parallel''' if

:<math>\nabla_{\dot\gamma(t)}X=0\text{ for }t \in I.\,</math>

In the case when <math>E</math> is the tangent bundle whereby <math>X</math> is a tangent vector field, this expression means that, for every <math>t</math> in the interval, tangent vectors in <math>X</math> are "constant" (the derivative vanishes) when an infinitesimal displacement from <math>\gamma(t)</math> in the direction of the tangent vector <math>\dot{\gamma}(t)</math> is done. 

Suppose we are given an element ''e''<sub>0</sub> &isin; ''E''<sub>''P''</sub> at ''P'' = ''&gamma;''(0) &isin; ''M'', rather than a section. The '''parallel transport''' of ''e''<sub>0</sub> along ''&gamma;'' is the extension of ''e''<sub>0</sub> to a parallel ''section'' ''X'' on ''&gamma;''.
More precisely, ''X'' is the unique part of ''E'' along ''&gamma;'' such that 
#<math>\nabla_{\dot\gamma} X = 0 </math>
#<math>X_{\gamma(0)} = e_0.</math>
Note that in any given coordinate patch, (1) defines an [[ordinary differential equation]], with the [[initial condition]] given by (2). Thus the [[Picard–Lindelöf theorem]] guarantees the existence and uniqueness of the solution.

Thus the connection &nabla; defines a way of moving elements of the fibers along a curve, and this provides [[linear isomorphism]]s between the fibers at points along the curve:
:<math>\Gamma(\gamma)_s^t : E_{\gamma(s)} \rightarrow E_{\gamma(t)}</math>
from the vector space lying over &gamma;(''s'') to that over &gamma;(''t''). This isomorphism is known as the '''parallel transport''' map associated to the curve. The isomorphisms between fibers obtained in this way will, in general, depend on the choice of the curve: if they do not, then parallel transport along every curve can be used to define parallel sections of ''E'' over all of ''M''. This is only possible if the '''[[Curvature form|curvature]]''' of &nabla; is zero.

In particular, parallel transport around a closed curve starting at a point ''x'' defines an [[automorphism]] of the tangent space at ''x'' which is not necessarily trivial. The parallel transport automorphisms defined by all closed curves based at ''x'' form a [[transformation group]] called the [[holonomy group]] of &nabla; at ''x''. There is a close relation between this group and the value of the curvature of &nabla; at ''x''; this is the content of the [[Holonomy#Ambrose–Singer theorem|Ambrose–Singer holonomy theorem]].

===Recovering the connection from the parallel transport===

Given a covariant derivative ∇, the parallel transport along a curve &gamma; is obtained by integrating the condition <math>\scriptstyle{\nabla_{\dot{\gamma}}=0}</math>. Conversely, if a suitable notion of parallel transport is available, then a corresponding connection can be obtained by differentiation. This approach is due, essentially, to {{harvtxt|Knebelman|1951}}; see {{harvtxt|Guggenheimer|1977}}. {{harvtxt|Lumiste|2001}} also adopts this approach.

Consider an assignment to each curve &gamma; in the manifold a collection of mappings
:<math>\Gamma(\gamma)_s^t : E_{\gamma(s)} \rightarrow E_{\gamma(t)}</math>
such that
# <math>\Gamma(\gamma)_s^s = Id</math>, the identity transformation of ''E''<sub>&gamma;(s)</sub>.
# <math>\Gamma(\gamma)_u^t\circ\Gamma(\gamma)_s^u = \Gamma(\gamma)_s^t.</math>
# The dependence of &Gamma; on &gamma;, ''s'', and ''t'' is "smooth."
The notion of smoothness in condition 3. is somewhat difficult to pin down (see the discussion below of parallel transport in fibre bundles). In particular, modern authors such as Kobayashi and Nomizu generally view the parallel transport of the connection as coming from a connection in some other sense, where smoothness is more easily expressed.

Nevertheless, given such a rule for parallel transport, it is possible to recover the associated infinitesimal connection in ''E'' as follows. Let &gamma; be a differentiable curve in ''M'' with initial point &gamma;(0) and initial tangent vector ''X'' = &gamma;&prime;(0). If ''V'' is a section of ''E'' over &gamma;, then let
:<math>\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - V_{\gamma(0)}}{h} = \left.\frac{d}{dt}\Gamma(\gamma)_t^0V_{\gamma(t)}\right|_{t=0}.</math>
This defines the associated infinitesimal connection &nabla; on ''E''. One recovers the same parallel transport &Gamma; from this infinitesimal connection.