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For a metric space, $(X,d)$, define the following:

  1. for every $x \in X$, the local similitude group, $Sim(X, x)$, is the set of all surjective similitudes $X \to X$ which fix $x$.
  2. for every $x \in X$, the local isometry group, $Iso(X, x)$, is the set of all surjective isometries $X \to X$ which fix $x$.
  3. the (global) isometry group, $Iso(X)$, is the group of all surjective isometries $X \to X$.
  4. an ang is a triple of points $(r_1, v, r_2)$. (The notation is meant to be suggestive, $r_1$ stands for "ray № 1", $v$ stands for "vertex", $r_2$ stands for "ray № 2", and the word has three letters because it denotes a triple of points).
  5. A point $b \in X$ is said to be between points $a,c \in X$ if and only if $$d(a,c) = d(a,b) + d(b,c) $$ i.e. if the triangle inequality becomes an equality.
  6. An angle is an equivalence class of angs under the following equivalence relation: $(r_1^*, v, r_2^*) \sim (\tilde{r_1}, v, \tilde{r_2})$ if and only if (1) [ $r_1^*$ is between $v$ and $\tilde{r_1}$, or $\tilde{r_1}$ is between $r_1^*$ and $v$]; and (2) [$r_2^*$ is between $v$ and $\tilde{r_2}$, or $\tilde{r_2}$ is between $r_2^*$ and $v$].

  7. an angle measure is a function from the set of all angles in $(X,d)$ to $\mathbb{R}$, which is (1) invariant under the action of the global isometry group, i.e. if two angles (have representatives which) are related by an isometry then they must have the same angle measure, (2) for angles with a given vertex $v$, the angle measure must be invariant under the action of $Sim(X,v)$.

  8. two angles are congruent if and only if they have the same angle measure.

In particular, note that two congruent angles need not be related by a surjective isometry, and two congruent angles need not be related by a surjective vertex-fixing similitude, even though the converses of these statements are automatically true by the definition of angle measure.

isotropic metric space - two angs with the same vertex $v$ are congruent if and only if they are related by the action of the local similitude group, $Sim(X,x)$.

This is an attempt to say, for general metric spaces, that "the notion of angle is rotationally invariant, i.e. does not depend on direction", even though we don't have a notion of rotation in general metric spaces. Basically, that the local isometry group at each point is "rich enough" (has enough "rotations") so as to be compatible with the angle measure.

<Examples> Riemannian manifolds, since (unlike for general Finslerian manifolds), the metric tensor tensor depends only on position, not position and direction. (See also.)

Finite-dimensional Banach spaces with norm induced by an inner product. <Examples>

We say that, given two angs $(r_1^*, v^*, r_2^*)$ and $(\tilde{r_1}, \tilde{v}, \tilde{r_2})$, they satisfy the SAS (Side-Angle-Side) criterion if and only if (1) they are congruent, i.e. they have the same angle measure $(r_1^*, v^*, r_2^*) \sim (\tilde{r_1}, \tilde{v}, \tilde{r_2})$, (2) $d(r_1^*, v^*) = d(\tilde{r_1}, \tilde{v})$, and (3) $d(r_2^*, v^*) = d(\tilde{r_2}, \tilde{v})$.

We say that a metric space $(X,d)$ satisfies the SAS (Side-Angle-Side) Postulate if and only if, given two angs satisfying the SAS criterion, they are related by the global isometry group.

Lemma: A metric space which satisfies the SAS Postulate is isotropic.

Proof: Still working on it -- the lemma might not be true with the given definitions, which would suggest that the definitions need to be changed/fixed.

Notes:
On definition 6:
Is this definition of angle compatible with already existing generalizations, e.g. (1)(2) ?

Also this definition of angle only seems to make sense when the notions of (1) betweenness and of (2) homothetic transformations/scaling/$Sim(X,x)$ coincide with each other. (I want to say this occurs when the metric space is an externally convex M-space.) However, that only occurs in really special spaces. Also it seems like this definition tries to make the notion of angle symmetric in $r_1$ and $r_2$, which we shouldn't necessarily expect to be the case (see page 4 here).

$\tilde{6}$ (alternate). An angle is an equivalence class of angs, with two angs equivalent if and only if they have the same vertex $v$ and are related by the local similitude group, i.e. if $d(v, r_1^*) = \lambda d(v, \tilde{r_1})$ and $d(v, r_2^*) = \lambda d(v, \tilde{r_2})$ for some $\lambda > 0$. Note: This definition would also make isotropic metric space below much easier to understand, allowing one to replace angs with angles.

On definition 7:
For really degenerate, "non-homogeneous", metric spaces, would it make more sense to create a separate angle measure function for each point, the same way there is a separate local isometry or similitude group for each point, and concomitantly drop the requirement that the angle measures have to be invariant under the global isometry group $Iso(X)$? The idea is that "angle measure" should be the "infinitesimal" and "scale-invariant" analog of "distance" for a metric space, so it's really only invariance under $Sim(X,v)$ at each $v$ which is most crucial for describing that.

Also, although it's necessary to define congruence of angles, the same way the metric is necessary to define congruence of pairs of points, there doesn't seem to be any "natural" choice of angle measure except in special cases where "circles" and their "circumferences" make sense.

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This would be a better solution:

Define an angle to be a triple of points $(r_1, v, r_2)$ in the metric space.

We say that angles $(r_1, v, r_2)$ and $(\rho_1, v, \rho_2)$ are related by the action of the local similitude group $Sim(X,v)$ if and only if there exists an $f \in Sim(X,v)$ such that $f(r_1) = \rho_1$ and $f(r_2) = \rho_2$.

We write this as $(r_1, v, r_2) \sim_v (\rho_1, v, \rho_2)$.

(Because of the implicit assumption that $f$ is surjective, this relationship is symmetric - just use $f^{-1}$. The $Id$ is in $Sim(X,v)$ so the relationship is also reflexive, and the fact that $Sim(X,v)$ is closed under composition means that the relationship is also transitive -- i.e. it's an equivalence relation.)

Denote the set of all angles with vertex $v \in X$ by $angle (X,v)$.

For each point $v \in X$, we define an angle measure to be any real-valued function $\measuredangle_v: angle(X,v) \to \mathbb{R}$ with the following constraint: if $(r_1, v, r_2)$ and $(\rho_1, v, \rho_2)$ are related by the action of $Sim(X,v)$, i.e. $(r_1, v, r_2) \sim_v (\rho_1, v, \rho_2)$, then $\measuredangle_v((r_1, v, r_2))=\measuredangle_v((\rho_1, v, \rho_2))$.

We say that two angles $(s_1, v, s_2)$ and $(\sigma_1, v, \sigma_2)$ are congruent if they have the same angle measure, written $(s_1, v, s_2) \cong_v (\sigma_1, v, \sigma_2)$, in other words $$(s_1, v, s_2) \cong_v (\sigma_1, v, \sigma_2) \iff \measuredangle_v((s_1, v, s_2)) = \measuredangle_v((\sigma_1, v, \sigma_2)) \,. $$

Note that $(r_1, v, r_2) \sim_v (\rho_1, v, \rho_2) \implies (r_1, v, r_2) \cong_v (\rho_1, v, \rho_2) \,, $ BUT $$(r_1, v, r_2) \cong_v (\rho_1, v, \rho_2) \kern.6em\not\kern -.6em \implies (r_1, v, r_2) \sim_v (\rho_1, v, \rho_2) \,.$$

We say that two angles $(r_1, v, r_2)$ and $(\rho_1, v, \rho_2)$ satisfy the SAS criterion if and only if: $$(r_1, v, r_2) \cong_v (\rho_1, v, \rho_2) \,, \quad d(r_1, v) = d(\rho_1, v) \,, \text{ and }d(r_2, v)=d(\rho_2, v) \,.$$

Then one can say that the angle measure $\measuredangle_v$ is isotropic at $v$ if and only if: two angles $(r_1, v, r_2)$ and $(\rho_1, v, \rho_2)$ satisfying the SAS criterion is equivalent to them being isometric to each other.

(I.e., two angles satisfy the SAS criterion if and only if they can be "rotated into each other", i.e. are related by the action of the local isometry group $Iso(X,x)$.)

With these definitions, the proof of the lemma above is immediate. Also, we can skip out on the ugly definitions of ang and angle from above and have fewer equivalence relations to deal with.

The only downside is that we have to define the extra data of an angle measure at every single point. While the data does have a compatibility requirement with the space's metric, there nevertheless (as defined) is no "natural" way to choose an angle measure at a point, so the relationship between the angle measure at each point and the metric is not canonical, but even more problematically, as a result there is no canonical relationship between the angle measures at different points (the way there would be if all of the angle measures were derived from the metric, which is common to each point of the metric space).

In practice though, this might be the best we can expect. While there is a canonical way to choose an angle measure for the Euclidean plane, even as soon as we do something simple, like change the norm we use for $\mathbb{R}^2$, ambiguity arises about whether we should continue to use the (isotropic) angle measure from Euclidean space, or if we should use a (non-isotropic) angle measure derived from the norm's unit ball. Anyway, my concern wasn't how much of the definition of angle measure for the Euclidean plane should I imitate, my concern was what properties does angle measure have in the Euclidean case which can be stated in the case of a completely general metric space without making any additional assumptions about the objects which exist on the space or what properties it satisfies, and this is what I came up with.

Still, it would be possible to go even further, by allowing $\measuredangle_v$ to map into any set, not just $\mathbb{R}$ (although ideally it would still be the case that $\measuredangle_v$ maps into the same set for all $v \in X$). (For some idea of what this might look like for a set other than $\mathbb{R}$, see here, although I am not sure whether the proposed definition of angle measure satisfies my single basic property, that two angles related by the action of the local similitude group possess the same angle measure.)

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