Consider a geodesic $c(t)$ in a smooth manifold $M$ connecting $p$ and $q$, $exp_p:T_pM \to M$ is a diffeomorphism when restricted to a neighborhood of $0$. Now suppose that $exp_p$ has no critical point in $B$ a ball around $0$ which contains $exp_p(c'(0))$, then it is a local diffeomorphism when restricted to this ball by inverse function theorem.
Now I want to show that I can lift $c$ to the tangent space.
Of course we can lift it locally by covering $c$ with diffeomorphic neighborhoods, but in the intersection of those neighborhoods, how do I show that they fit with each other well?
It basically follows from the definition of $\exp_p$. The curves $c(t)$ and $c_1(t) = \exp_p (tv)$, where $v =c'(0) \in T_pM$, are both geodesics and $$c(0) = c_1(0),\ \ c'(0) = c'_1(0).$$ Thus $c(t) = c_1(t)= \exp_p(tv)$ whenever both sides are defined. This of course just say $c$ is lifted to the tangent plance $T_pM$ (to the line $tv$).