Question: What is the correct definition of "rotation" in a general metric space?
Is the following correct?
Let $(X,d)$ be a metric space. Let $G_x$ be the group of isometries of $X$ which fix the point $x \in X$. Then any isometry in $G_x$ is a "rotation about $x$".
Also note that, if $X$ is an orientable vector space, I am apathetic or agnostic about whether rotations should be allowed to reverse orientation or if they must preserve orientation.
A definition for the full generality of arbitrary metric spaces would be preferred, but one valid only for, say, Finsler manifolds or even just (finite-dimensional) normed spaces would also be appreciated, since it would still answer my previous question.
Context: My working assumption right now is that, for a vector space, it is the group of isometries which fix the origin, see my previous question to which this is a follow-up.
This question on MathOverflow seems like it might be using this definition. However, I don't understand what is meant by "isotropic space" -- it seems more general than isotropic manifold. The transitivity seems related to a theorem in Spivak's Comprehensive Introduction to Differential Geometry which I have mentioned in two previous questions (1)(2). Also this answer on Math.SE repeats the claim that "unit balls with respect to other norms are not rotationally invariant", which, as I pointed out in my previous question is either trivial or insightful depending upon one's definition of "rotation", which seems to never be specified in this instance.
The answer seems like it might also have something to do with CAT(0) spaces, since CAT(0) spaces apparently satisfy a sort of parallelogram law, and a norm is induced by an inner product if and only if it satisfies the parallelogram law.
However, the parallelogram law is equivalent (I think) to the Pythagorean theorem (at least for $L^p$ norms) and the Pythagorean theorem is equivalent to the parallel postulate and a bunch of other conditions, at least for Euclidean space. (See also this question.) But one of these formulations seems to have more to do with geodesic completeness than any obvious notion of angle, see my previous question. This might be true since CAT(0) spaces have unique geodesics, see this answer.
It is often said that an inner product "induces notions of length and angle" and, as far as I can tell, at least in simple spaces the notions of rotation and angle are intimately related. (Although for arbitrary metric spaces angles seem to have more to do with equivalence classes of triples of points (see also Section 3.6.5. here) and similarity transformations than point-fixing isometries.)
At the very least, the part of inner product which corresponds to both its notion of angle and its corresponding orthogonal group (group of rotations?) is the conformal structure it belongs to. However, conformal structures do not seem to be interesting objects of study (see my previous question) which suggests that they are not actually important in defining a notion of rotation.
This answer mentions
compact one-parameter groups of "rotations" without elaborating.
All of these questions by @Asaf Shachar are of interest: (1)(2)(3)(4)(5). Reading them either taught me or confirmed for me (I don't remember) that the orthogonal group (and thus a notion of rotation?) is only unique up to a scalar multiple for an inner product.
TL;DR Context: I don't understand what is "fundamental" about the notion of rotation even for the simplest example of rotations: the orthogonal group on Euclidean space. (See also: ) Is it the inner product up to scalar multiple? The parallelogram law? The parallel postulate? Unique geodesics? The Cat(0) inequality? Point-fixing isometries? etc.
Thus I am very uncertain of how to generalize the notion of rotation from Euclidean space to arbitrary metric spaces -- the orthogonal group has so much structure, it is hard for me to tell which part of the structure is "essential" for codifying the "notion of rotation".