I was wondering if this is a legitimate way to define the induced basis of the tangent space in normal coordinates.
So the exponential map is a diffemorphism $exp:U \subset T_pM \rightarrow V \subset M.$ Then I assume that $\partial_i|_{exp(tR)}= Dexp(tR)(e_i)$ where $(e_i)$ is a fixed basis of the tangent space and $t \mapsto tR$ is a line in $U.$
Now my question is: Does this really define the induced basis of $T_{exp(tR)}M$?