(Am writing this post in a rush, out of office, so cannot give adequate links etc right now.)

There is a classical and well-understood definition of what it means for a continuous function $f:[a,b]\to{\mathbb R}$ to be absolutely continuous: we should have $$ \sup \{ \sum_{i=1}^n (t_i-t_{i-1}) \vert f(t_i)-f(t_{i-1}) \vert \} < \infty $$ where the supremum is over all partitions $a=t_0 < t_1 < \dots < t_n=b$.

From my limited reading on Wikipedia and various other online searches, it transpires that there are several different notions of "functions of bounded variation" defined on higher-dimensional "domains", with the quotes indicating that I am uncertain of the precise technical qualifiers. (The setting given in Wikipedia is for open subsets of ${\mathbb R}^n$; googling has also shown links to papers defining BV for Riemannian manifolds.)

It is not clear to me how these definitions in higher dimensions could be approached in a naive way via approximating sums, as in one variable. So my rather naive question is: can such a definition be made to work on, say, the $2$-sphere? What about higher spheres?

Also, since BV/AC are metric notions and not just topological ones, would I get definitions closer to the one-dimensional case if I worked with a suitable polyhedron or a triangulation of the $2$-sphere? (and if so, does such an approach obviously generalise to higher spheres?)

(The motivation for this question is slightly indirect, and I may try to give more context when I am next in the office with free time and working brain.)