Let $\nabla\colon\Gamma(TM)\times\Gamma(E)\to\Gamma(E),(X,\sigma)\mapsto\nabla_X\sigma$ be a connection on a vector bundle $E$ over a smooth manifold $M$, and let $\gamma\in C^\infty(I,M),I:=[0,1]$ be a smooth curve in $M$. In our lecture we "defined" the parallel transport of $v_0\in E_{\gamma(0)}$ along $\gamma$ as follows.
The equation $\nabla_{\dot\gamma}\sigma\equiv0$ corresponds to an ODE which for any given initial value $\sigma_{\gamma(0)}=v_0\in E_{\gamma(0)}$ has a unique solution $\sigma\in\Gamma(E)$. We call $\sigma_{\gamma(1)}\in E_{\gamma(1)}$ the parallel transport of $v_0$ along $\gamma$.
I see one problem with this definition: The derivative $\dot\gamma=D\gamma(\partial_t)\colon I\to TM$ is not a vector field on $M$, i.e. it is not a section of $TM$. Even when viewed as a map $\gamma(I)\subset M\to M$, it isn't defined on all of $M$ making it unsuitable as an argument to $\nabla$. I am aware of the locality (or tensoriality) of the connection. But if there does not exist any vector field $X\in\Gamma(TM)$ which coincides with $\dot\gamma\colon\gamma(I)\to TM$ on $\gamma(I)$, then you can't use locality to argue that $\nabla_{\dot\gamma}\sigma$ is well-defined. If $\dot\gamma\colon\gamma(I)\to TM$ can actually be extended to a full section $\dot\gamma\in\Gamma(TM)$, then problem solved. But if $\gamma$ intersects itself with different velocities at the intersection, extending $\dot\gamma$ becomes impossible.
The only reasonable approach I could come up with is to define some pullback connection $\gamma^*\nabla\colon\Gamma(TI)\times\Gamma(\gamma^*E)\to\Gamma(\gamma^*E)$ on the pullback bundle $(\gamma^*E)_t=T_{\gamma(t)}M$ (over $I$) and define $\sigma\in\gamma^*E$ to be the solution of $(\gamma^*\nabla)_{\partial_t}\sigma\equiv0$ with initial value $\sigma_0=v_0\in T_{\gamma(0)}M$.
After a little bit of digging I found this Wikipedia article, where the author claims that there exists a unique connection $\gamma^*\nabla$ on $\gamma^*E$ satisfying $(\gamma^*\nabla)_X(\gamma^*\sigma)=\gamma^*(\nabla_{d\gamma(X)}\sigma)$ for all $X\in^?\Gamma(TI)$ and $\sigma\in\Gamma(E)$. But this seems to suffer the same problem as above, because $d\gamma(X)\colon I\to TM$ is not a section of $TM$ leaving $\nabla_{d\gamma(X)}\sigma$ undefined.
The same problem occurs in the definition of geodesics.
$\gamma$ is called a geodesic of $M$ with respect to an affine connection $\nabla\colon\Gamma(TM)\times\Gamma(TM)\to\Gamma(TM)$, if $\nabla_{\dot\gamma}\dot\gamma\equiv0$.
But again, $\dot\gamma\notin\Gamma(TM)$.
This brings me to my question: How are geodesics and the parallel transport of a vector $v_0\in E_{\gamma(0)}$ actually defined? Is there a rigorous way of defining these concepts?
Best Answer
The formally correct would be indeed to use the pullback connection, which exists by the following theorem:
Note that since a connection is tensorial tensorial in the first slot it actually makes sense to plug in tangent vectors instead of vectorfields. The output will then be also just a tangent vector.
Now if $\nabla$ is a connection on $TM$ and $\gamma: I\to M$ is a smooth curve then $\dot\gamma\in\Gamma(\gamma^*TM)$, so if $\frac{\partial}{\partial s}$ is the standard vectorfield on $I$ (meaning the vectorfield coming from the curve $t\mapsto t$) then $\ddot \gamma(t)=(\gamma^*\nabla_{\frac{\partial}{\partial s}}\dot\gamma)(t)$ is well defined and may be also denoted by $(\nabla_{\dot\gamma}\dot\gamma)(t)$.
As a last word: If the curve $\gamma$ satisfies $\dot\gamma(t)\neq 0$ for all $t\in I$ then one can avoid using the pullback connection (even if $\gamma$ has self-intersections) by using the locality priciple and an extension argument, see for example here.