Recovering the connection from the parallel transport

Given a covariant derivative ∇, the parallel transport along a curve γ is obtained by integrating the condition ${\displaystyle \scriptstyle {\nabla _{\dot {\gamma }}=0}}$. 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 Knebelman (1951); see Guggenheimer (1977). Lumiste (2001) also adopts this approach.

Consider an assignment to each curve γ in the manifold a collection of mappings

${\displaystyle \Gamma (\gamma )_{s}^{t}:E_{\gamma (s)}\rightarrow E_{\gamma (t)}}$

such that

1. ${\displaystyle \Gamma (\gamma )_{s}^{s}=Id}$, the identity transformation of E_γ(s).
2. ${\displaystyle \Gamma (\gamma )_{u}^{t}\circ \Gamma (\gamma )_{s}^{u}=\Gamma (\gamma )_{s}^{t}.}$
3. The dependence of Γ on γ, 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 γ be a differentiable curve in M with initial point γ(0) and initial tangent vector X = γ′(0). If V is a section of E over γ, then let

${\displaystyle \nabla _{X}V=\lim _{h\to 0}{\frac {\Gamma (\gamma )_{h}^{0}V_{\gamma (h)}-V_{\gamma (0)}}{h}}=\left.{\frac {d}{dt}}\Gamma (\gamma )_{t}^{0}V_{\gamma (t)}\right|_{t=0}.}$

This defines the associated infinitesimal connection ∇ on E. One recovers the same parallel transport Γ from this infinitesimal connection.

Special case: the tangent bundle

Let M be a smooth manifold. Then a connection on the tangent bundle of M, called an affine connection, distinguishes a class of curves called (affine) geodesics.^[2] A smooth curve γ: I → M is an affine geodesic if ${\displaystyle {\dot {\gamma }}}$ is parallel transported along ${\displaystyle \gamma }$, that is

${\displaystyle \Gamma (\gamma )_{s}^{t}{\dot {\gamma }}(s)={\dot {\gamma }}(t).\,}$

Taking the derivative with respect to time, this takes the more familiar form

${\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}=0.\,}$

Geodesics

If ∇ is a metric connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves. More precisely, first note that if γ: I → M, where I is an open interval, is a geodesic, then the norm of ${\displaystyle {\dot {\gamma }}}$ is constant on I. Indeed,

${\displaystyle {\frac {d}{dt}}\langle {\dot {\gamma }}(t),{\dot {\gamma }}(t)\rangle =2\langle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t),{\dot {\gamma }}(t)\rangle =0.}$

It follows from an application of Gauss's lemma that if A is the norm of ${\displaystyle {\dot {\gamma }}(t)}$ then the distance, induced by the metric, between two close enough points on the curve γ, say γ(t₁) and γ(t₂), is given by

${\displaystyle {\mbox{dist}}{\big (}\gamma (t_{1}),\gamma (t_{2}){\big )}=A|t_{1}-t_{2}|.}$

The formula above might not be true for points which are not close enough since the geodesic might for example wrap around the manifold (e.g. on a sphere).