Axiomatization of angle measuring in real vector spaces
In linear algebra / analytic geometry it is common to define the angle between two vectors $u,v \in V$ of an euclidean vector space $V$ by $\angle (u,v) := \arccos \frac{(u,v)}{\|u\| \cdot \|v\|}$. Is there a general theory analogous to measure theory that abstracts from this concrete mapping by allowing an angle measure to be any function which satisfies the most important properties of the above one?
I am thinking of a definition like this:
Let $V$ be a real vector space. A mapping $\angle : V \times V \to \mathbb{R}$ is called an angle measure, if it satisfies the following axioms for all $u,v \in V$ and $\lambda, \mu \in \mathbb{R}^+$:
- $\angle(u,v) = \angle (v,u)$
- $\angle(\lambda u, \mu v) = \angle(u,v)$
- $\angle(u,w) + \angle(w,v) = \angle(u,v)$ where $w = \lambda u + \mu v$
- $\angle(u,v) \leq \angle(u,w) + \angle(w,v)$ for all $w$
- ...
Is there a finite set of such axioms that characterizes the usual angle measures coming from inner products (in finite dimensional spaces)? Are there interesting examples of angle measures coming not from inner products?
Here is a detailed version of my comments.
I will be working under the assumption that $V$ is finite dimensional, of dimension $n$. Let $S^{n-1}$ denote the standard unit sphere in $V$.
First, notice that the function $\angle$ induces a metric on the sphere $S=S^{n-1}$. Due to your third condition, this metric has the property that great circles in $S$ are geodesics for this metric. Such metrics are called "projective", I think. Conversely, given a projective metric $\angle$ on $S$, one extends it to $V$ by the formula $$ \angle(tu, s v)=\angle(u, v) $$ where $u, v\in S$ and $s, t>0$. The resulting angle function on $V$ will satisfy all your requirements.
I will add two conditions to the projective metric $\angle$:
$\angle$ defines the standard topology on $S$ (this condition can be relaxed, but let's not go into it).
If $u, v, w\in S$ do not belong to a great circle (equivalently, to a common 2-dimensional subspace) then $$ \angle(u,v)+ \angle(v,w)> \angle(u,w). $$
Projective metrics $\angle$ satisfying these 2 extra conditions are called "Desarguesian".
Note that one can add one extra (and quite reasonable) requirement that $$ \angle(u,v)=\angle(-u, -v). $$ In other words, $\angle$ descends to a metric on the projective space $RP^{n-1}$.
Hilbert in his 4th problem asked for a classification of Desargusuan metrics (he was allowing metrics on more general domains than $S$, line plane, but he was primarily interested in the case when $S$ has dimension 2 or 3).
There are two very nice survey articles which summarize history of Hilbert's 4th problem: here and here.
In short: Busemann found a very general way to construct Desargusuan metrics on sphere (and more general spaces, line real-projective space and convex domains in it) using continuous measures on the space of (linear) hyperplanes in $V$. More specifically, Busemann defined $\angle(u, v), u, v\in S$, as the total measure of hyperplanes separating $u$ from $v$.
Pogorelov proved that all Desargusuan metrics (subject to some mild regularity assumption in higher dimensions) appear from Busemann's construction. (Ambartzumian gave an alternative proof in the case when $n=3$.) To the best of my knowledge, Pogorelov's proof is regarded as a solution of Hilbert's 4th problem. There are many other results and open problems in this field, you should read the linked survey articles for more details.