Suppose you have a curve $\alpha$ in a manifold. You are at a point $\alpha(t)$ of that curve. The curvature of $\alpha(t)$ is the same as the curvature of the curve $exp^{-1}_{\alpha(t)}(\alpha(s))$ at $s=t$. The previous curve is what you see from the tangent space of $\alpha(t)$.

Now imagine you move to another point $x$, and want to determine how the curvature of $\alpha(t)$ has changed when viewed from the tangent space of $x$.

That is, which is the curvature of $exp^{-1}_{x}(\alpha(s))$ at $s=t$?

It is fine for me for now to know the answer when the manifold is an space of constant sectional curvature. I suppose that in such a case it will only depend on the distance between $x$ and $\alpha(t)$ and the relative position between the geodesic joining $x$ and $\alpha(t)$ with respect to the hyperplane (in the tangent space of $\alpha(t)$) tangent to $\alpha$ at $t$.