Let $(M,g)$ be a Riemannian manifold, and let $\{(p_i,y_i):p_i \in M, y_i \in \mathbb{R}\}$ be a collection of real values assigned to distinct points of $M$. Is there any good coordinate independent way to interpolate and find a function $y:M\rightarrow \mathbb{R}$ such that $y(p_i) = y_i$?
When $(M,g)$ is flat and $p_i$ gives a triangulation, we can interpolate linearly on each triangle (unambiguously, using the metric), giving us a piecewise-linear $y(p)$ on the union of the triangles. For example, on $M = \mathbb{R}$ this is just piecewise linear interpolation. The same procedure works on $\mathbb{R}^n$. What if instead $M$ was the sphere? The "linear" requirement probably doesn't make sense, but is there some weaker condition we can have?