The length $\gamma : [a,b] \rightarrow M$ are defined by integrating their tangent vectors along $\gamma$ a Riemannian manifold.
$l_{\gamma} := \int_{a}^{b} |\gamma^{'}|dt$
Now I have the question that it's said that the tangent at a point is not unique, then how is this integration well defined or computed.
Now I may be confusing between that there is no unique tangent at a point in manifold Vs tangent of curves. Maybe curves have unique tangents depending on the parameterization, but more clarity be helpful.