TITLE: How "should" I define "absolutely continuous" functions on e.g. n-spheres?
QUESTION [6 upvotes]: (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.)

REPLY [4 votes]: There is a difference  between  BV and AC.            AC functions on an interval are  integrable functions  whose  distributional derivatives are   representable by integrable functions.  BV function on an interval  are  integrable functions whose  distributional derivatives are distributions representable  by (signed) measures.
We can then  define  the space  AC functions on a Riemann manifold to be the Sobolev space $W^{1,1}$ (one derivative in $L^1$), i.e.,  integrable  functions with integrable gradients.   These functions are definitely not continuous  so they may not be as appealing to you.