What is the definition of "rotation" in a general metric space? (Or a Finsler manifold?)

Question

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: [1][2]) 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".

Answer 1

Mathematical words often get re-used in contexts where there is only an analogy rather than a precise mathematical principle covering precisely all the cases, old and new. This, I think, is what you are encountering with the terminology "rotation". It's a mistake to over-interpret what is going on in with the word "rotation" in each new situation.

Here, for example, is one way that the term "rotation" might grow in use through analogy.

We can certainly define "rotations of the Euclidean plane" with precision. For example:

1. Define $f : \mathbb{R}^2 \to \mathbb{R}^2$ to be a rotation if there exists $\theta \in (0,2\pi)$ and $a,b \in \mathbb{R}$ such that $$f(x,y) = (x \cos \theta + y \sin \theta, - x \sin \theta + y \cos(\theta)) + (a,b) $$

Then we can prove theorems about rotations, for example:

2. $f : \mathbb{R}^2 \to \mathbb{R}^2$ is a rotation if and only if $f$ is an orientation preserving isometry with a unique fixed point.

Now, suppose we are studying the orientation preserving isometries of coordinate Euclidean 3-space $\mathbb{R}^3$. We discover, much to our consternation, that none of them have a unique fixed point.

So, do we shrug our shoulders and say "Euclidean 3-space has no rotations"?

Perhaps, although that feels uncomfortable given our real-world experience with 3-dimensional rotations.

There is, however, another possibility, namely to change our analogy by proving that 1 and 2 are equivalent to

3. $f : \mathbb{R}^2 \to \mathbb{R}^2$ is a rotation if and only if it is an orientation preserving isometry and its fixed point set is a codimension 2 affine subspace.

And now, perhaps, we are happy, because this indeed generalizes very nicely to Euclidean spaces of all dimensions. And perhaps now we think know what rotations are.

Well, maybe. Every time we go down roads of further and further generalization, we may be tempted to stretch the analogy further and make some grand generalized "rotation" definition. But somehow that misses the point.

The real point is: We should study isometries of $\mathbb{R}^2$, or $\mathbb{R}^3$, or $\mathbb{R}^n$, or CAT(0) Riemannian metric spaces, or whatever new context we encounter, for what they really are. If analogy with rotations of $\mathbb{R}^2$ help us understand isometries in these new contexts, by all means reuse the terminology. If it is misleading, or if it just is unhelpful, then don't use the terminology. At the very least, don't stretch the analogy beyond the breaking point.

Answer 2

There is no widely used definition of rotation in a general metric space. This becomes clear after Googling a bit and reading the comments and answers of this question and your previous question. Therefore I would say there is no such thing as the definition.

You can define anything you like as long as it makes some sense. Your definition of $G_x$ definitely makes sense, since when $X = \mathbb{R}^n$ and $d$ is the Euclidean norm, the group $G_x$ is equal to the orthogonal group $O(n)$.

Generalizing proper rotations

The group $G_x$ generalizes both proper and improper rotations. The word rotation sometimes includes improper rotations and sometimes it does not. I have thought about how to generalize proper rotations to arbitrary metric spaces.

What's the difference between proper and improper rotations? I would say that a proper rotation is continuous, in the sense that you can move the elements of the metric space by a lot of small bits in order to obtain the complete rotation, while preserving the distance at all times.

You could use $$ \begin{array}{} P_x &=& \{f \in G_x \ |\ \exists (p \in [0,1] \to G_x): p(0) = \text{id}_X \land p(1) = f\ \land \\ && \qquad \forall(\alpha_1 \in [0, 1], y \in X, \epsilon > 0): \exists(\delta > 0): \forall(\alpha_2 \in [0,1]): \\ && \qquad \qquad |\alpha_1 - \alpha_2| < \delta \to d(p(\alpha_1)(y), p(\alpha_2)(y)) < \epsilon \\ &&\}, \end{array} $$ which is a subgroup of $G_x$, to generalize proper rotations to general metric spaces. For $\mathbb{R}^n$ with the Euclidean metric, the group is equal to the special orthogonal group $SO(n)$.

For $\mathbb{R}^n$ with the taxicab metric, $P_x$ will be the trivial group, since no continuous rotations are possible with the taxicab metric, only rotations of 90 degrees.

Answer 3

For a general metric space I would accept your definition. It is standard in metric vector spaces.

For a metric vector space a rotation is defined as an isometry which keeps at least one point fixed. Normally the origin as taken as the fixed point. If the metric is generated by a norm the Mazur-Ulam theorem shows the rotation is linear. In a vector space with an inner product the above definition gives the usual meaning, although it includes reflections among the rotations.