From this notes of B.K.Diver he introduce the Cartan's Development map, which can be understood as a rolling map. However, I am a bit confused with the theorem 4.1, which is stated as follows.
Given $M$ a compact Riemannian manifold, a smooth path $b$ in $T_oM$ with $b(0)=o$(the origin), there exist a smooth path $\sigma$ in $M$ such that$$\sigma'(s):=(\sigma(s),\frac{d\sigma(s)}{ds})=//_s(\sigma)b'(s)$$ where $ //_s(\sigma)v_0=(0,\Gamma_o^s(\sigma)v_0)$, and $\Gamma_o^s(\sigma)$ is the parallel transport from $T_oM$ to $T_{\sigma(s)}M$ along the curve $\sigma$.
Question:
- I dont understand the map $//_s(\sigma)$. Why are fixing the first argument at 0 and the differential equation stated above can never fufilled unless $\sigma(s)=0$. In my opinion we have to move $b'(s)$ to the origin and redefine $//_s(\sigma)v_0$ as $(\sigma(s),\Gamma_0^sv_0)$. Is this right?
- I don't see how it recovers a rolling map. By the differential equation given above, we have only that the $\sigma(s)$ is tangent to some transformation of $b(s)$ induced by the parallel transport. But for a rolling map we also need to require $\sigma(s)=g*b(s)$, where g is the induced transformation by the parallel transport. How can I see that this is given? Has it something to do that the smooth path $b$ is defined on the tangent space $T_oM$?