Algorithm 3 Covariant integration of $\frac{d\alpha }{dt}$
1:	Let w(0) = 0.
2:	for τ = 1 to k do
3:	$w\left(\frac{\tau }{k}\right)=\Pi \left(w\left(\frac{\tau -1}{k}\right);\alpha \left(\frac{\tau -1}{k}\right),\alpha \left(\frac{\tau }{k}\right)\right)+\frac{1}{k}\frac{d\alpha }{dt}\left(\frac{\tau }{k}\right)$
4:	Project $w\left(\frac{\tau }{k}\right)$ into $T_{\alpha \left(\frac{\tau }{k}\right)}(\mathcal{C})$ using Algorithm 2.
5:	end for