I'm having a hard time trying to understand something that I'm suspicious is pretty stupid. I'll refer to Wikipedia to settle the term's I'll refer to.
http://en.wikipedia.org/wiki/Normal_coordinates#Geodesic_normal_coordinates
That said, having $V=(X_1,...,X_n)\in T_p M$, how can the element $\gamma_V(t)=(tX_1,...,tX_n)$ be on the manifold $M$? Isn't the geodesic defined by $\gamma_V(t)=exp_p (t.V)$? This to me looks like just a parameterization of a straight line which does not need to be on the manifold, necessarily.
I need some help. I'll also link this lecture note which I'm trying to understand: http://www-personal.umich.edu/~wangzuoq/635W12/Notes/Lec%2023.pdf.
Thanks!
Linear coordinates $X_{i}$ on $T_{p}M$ are identified, via the exponential map, with normal coordinates in a sufficiently small neighborhood of $p$ in $M$. When one writes $\gamma_{V}(t) = (tX_{1},\dots tX_{n})$ in normal coordinates (as on the wikipedia page you linked), this identification is implicit.