Say that I want to do local averaging on a circle.
I have values for angles $\phi \in [0,1]$ where 1 "spins around" to 0. Let us call it $c[0,1]$
If I just do normal averaging $$\frac{1}{N}\sum_{i=1}^N \phi_i$$
I will get trouble because of the cut at 0,1.
I can instead remap $\phi \to \exp(2\pi i \phi)$ and change sum to product: $$\prod_{i=1}^{N} \exp(2\pi i \phi_i/N)$$
This will create a smooth behaviour around the angular jump.
How can I generalize this to more complicated surfaces ?
Own work:
Say for example I have an ordered pair on $\{c[0,1],\mathbb R^+\}$ which I want to smooth.
We can use the same construction above but encode the $\mathbb R^+$ into the radius part of the complex number. But then our complex numbers are kind of "exhausted" when comes to degrees of freedom of what to be able to represent.
Not an answer, too long for a comment.
tl;dr To get a useful answer I think you will have to edit the question to provide more context. I don't understand your "Say for example" example.
To compute averages on a surface (or manifold of any dimension) you need a way to measure areas on that surface. Your question seems to assume some kind of underlying "uniform distribution".
Your "average angle" example is in fact more subtle than you realize. See
Average of angles
and the wikipedia page referred to there.