Parallel transport and holonomy

If $\gamma : [0, 1] \to M$ is a path and $\nabla$ a connection we can consider the notion of moving a vector in $L_{\gamma(0)}$ to $L_{\gamma (1)}$ without changing it, that is parallel transporting a vector from $L_{\gamma (0)}, L_{\gamma (1)}$. Here change is measured relative to $\nabla$ so if $\xi(t) \in L_{\gamma (t)}$ is moving without changing it must satisfy the differential equation:

$$\nabla_{\dot\gamma} \xi = 0$$ (2.3)

where $\dot\gamma$ is the tangent vector field to the curve $\gamma$. Assume for the moment that the image of $\gamma$ is inside an open set $U_\alpha$ over which $L$ has a nowhere vanishing section $s_\alpha$. Then using (2.3) and letting $\xi(t) = \xi_{\alpha} (t) s_{\alpha} (\gamma (t))$ we deduce that

$${\frac {d \xi_{\alpha }} {dt}} = - A_{\alpha} (\gamma) \xi_{\alpha }$$

or

$$\xi_{\alpha }(t) = \exp \bigl(- \int_{0}^{t} A_{\alpha} (\gamma(t)\bigr) \xi_{\alpha }(0)$$ (2.4)

This is an ordinary differential equation so standard existence and uniqueness theorems tell us that parallel transport defines an isomorphism $L_{\gamma (0)} \cong L_{\gamma (t) }$. Moreover if we choose a curve not inside a special open set like $U_\alpha$ we can still cover it by such open sets and deduce that the parallel transport

$$P_{\gamma} : L_{\gamma (0)} \to L_{\gamma (1)}$$

is an isomorphism. In general $P_{\gamma}$ is dependent on $\gamma$ and $\nabla$. The most notable example is to take $\gamma$ a loop that is $\gamma (0) = \gamma (1)$. Then we define hol$(\gamma, \nabla)$, the holonomy of the connection $\nabla$ along the curve $\gamma$ by taking any $s \in L_{\gamma (0)}$ and defining

$$P_{\gamma} (s) =$$   hol$$\ (\gamma, \nabla). s$$

Example 2.5   A little thought shows that $\nabla$ on the two sphere preserves lengths and angles, it corresponds to moving a vector so that the rate of change is normal. If we consider the `loop' in Figure 4 then we have drawn parallel transport of a vector and the holonomy is $\exp (i\theta)$.

Figure 4: Parallel transport on the two sphere.