Consider some compact Riemannian manifold $M$. Fix some point $p$. Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$. Consider growth function $g(r)$ to be volume of such submanifold.
Question: What are the compact manifolds/(other metric spaces) such that growth is well approximated by the Gaussian normal distribution ?
For example is it true for homogeneous spaces for compact groups of large enough dimension ? For submanifolds in such spaces defined by algebraic equations ? I.e. hypersurfaces in $CP^{n}$ with respect to standard Fubini-Study metric ? Is it related to positive curvature ?
Example: Consider $M$ to be standard d-dimensional sphere $S^{d}$, take d=20 for example , growth is proportional $sin(r)^{d-1}$ which is well approximated by the Gaussian - see figure below. Examples for other $d$ can be found in : https://www.kaggle.com/code/alexandervc/gaussian-for-spheres (Thanks to Leonid Petrov for suggestion)
Another examples can be tori $T^d$, actually not only with Euclidian metric, but also for $L_1$-metric (Manhattan distance) - which can be seen as limit metric for the lattice distance (lattice step tends to zero).
Motivation: Question is related to the previous question for growth of FINITE NILPOTENT groups where we can also expect growth (in the sense of group theory) can also be well approximated by Gaussian in some cases. And the machine-learning perspective of the "embeddings" for graphs (Cayley graphs in particular). That is - we may hope metric on Cayley graph (just the shortest path length) should agree to some extent with Euclidian metric for "good" embeddings. Thus the Gaussian approximation property for growth in both cases should hopefully agree. (More on that perspective - see question - Permutohedron is expected to be "good" embedding of certain Cayley graph of $S_n$, which growth is quite Gaussian (see e.g. figure in that question ) and can be checked for Permutohedron embedding see figures (thanks to EUGENE DURYMANOV) ).
Why may work: roughly speaking when we calculate distance from $p1$ to $p2$ in higher dimensional manifold is kind of combined distances in each coordinate, for large number of coordinates central limit theorem should start working and we might get Gaussian.
PS
The question might be seen as kind of "metric central limit theorem", in addition to "topological central limit theorem" , and "group-theoretic central limit theorem". In all cases we see that result in question is combination of multiple components and central limit phenomena starts working. We can also think of "motivic" version - where motive of the flag manifold plays a role of the Gaussian.
