I am teaching myself some elementary differential geometry and am stuck on the concept of the tangent vector field of a smooth curve. I have searched the web for an hour or so but cannot find anything pertaining specifically to my issue here.
For a smooth curve $\gamma:(0,1)\rightarrow M$, where $M$ is a manifold with a connection $\nabla$, we can define the notion of autoparallel transport. We say the curve is autoparallely transported if $\nabla_{\dot{\gamma}}\dot{\gamma} = 0$ on the curve, where $\dot{\gamma}$ is the tangent vector field to $\gamma$.
My question is, how do we define this guy in terms of an element in the section of the tangent bundle? This is my initial guess, sort of with an abuse of notation:
\begin{equation} \dot{\gamma}(\bullet) := \dot{\gamma}(\gamma(\bullet)) = \dot{\gamma}\circ \gamma : (0,1)\to TM ,\end{equation}
which I think would make the latter $\dot{\gamma}\in\Gamma(TM)$, i.e. a vector field sort of sending $M\lvert_{\gamma}\to TM$. But something feels off, is this enough to be able to continue?
Best Answer
The condition is indeed what you want: $\forall t\in (0,1),\gamma'(t)\in T_{\gamma(t)}M$. More formally, you can look at the definition of a pullback bundle: here you are actually defining a section of the bundle $\gamma^*TM$.