Editing Parallel transport (section)

===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.