Axioms for angular or conformal structure Let $V$ be a real vector space. Is there a way to (directly) axiomatise the notion of a map $\Theta: V \times V \to \mathbb{R}$ being a measure of the angles between vectors? If we have an inner product $\langle , \rangle$ then such a map is naturally induced as
$$\Theta(u, v) = \cos^{-1} \left( \frac{\langle u, v \rangle}{|u| |v|} \right) $$
where $| \cdot |$ is the standard norm ($|u| = \sqrt{\langle u, u\rangle}$). But of course, two inner products could induce the same angles (certainly if, and I believe only if, one is a rescaling of the other). So is there a way to define directly what it is for something to be an angular measure? Some properties are obvious, like $\Theta(u, u) = 0$; I'm wondering if there's a generally accepted list of such properties.
By way of context, I'm ultimately hoping for a way of defining conformal structure that's more intrinsic than the usual "conformal structure is an equivalence class of metrics" definition. So I was thinking that a plausible candidate might be to define a conformal structure as a smooth assignment of an angular measure to each tangent space.
 A: This is an extended remark, not a full answer.
My experience in differential geometry suggests that the concept of angle isn't all that useful.
In practice it seems to be better to think of inner products modulo a scalar multiplier.
Angles are good concepts for thinking visually, but rarely for computations in conformal geometry.
The question is still interesting, though.
Your angle function $\Theta$ should be defined on $V^*\times V^*$, where $V^*=V\setminus0$ is the punctured space (not the dual).
Notice that the angle is invariant under positive scaling: $\Theta(\lambda u,\mu v)=\Theta(u,v)$ for $u,v\in V^*$ and $\lambda,\mu>0$.
To make use of this, let us say that $u,v\in V^*$ are in relation $u\sim v$ iff $u=\lambda v$ for some $\lambda>0$.
This is an equivalence relation, and the value of $\Theta(u,v)$ only depends on the equivalence classes of $u$ and $v$.
Therefore it makes sense to define the angle function on the quotient space $S=V^*/{\sim}$.
Let us call this new function $\Omega:S\times S\to\mathbb R$.
It satisfies $\Omega([u],[v])=\Theta(u,v)$ for all $u,v\in V^*$, where the brackets stand for equivalence classes.
The quotient space $S$ is a "sphere".
I use quotes, since it is only an abstract quotient space, but if you have a norm, every class has a natural unique representative (the unit vector) and the "sphere" can be identified with the metric sphere.
I specifically want to avoid fixing a norm or an inner product.
The angle function $\Omega$ is a metric on the sphere $S$.
This gives a lot of structure, but is still insufficient to characterize angle functions.
There are many additional properties, for example:


*

*The function $\Omega$ only takes values in $[0,\pi]$.

*There are the symmetries $\Omega(u,v)=\Omega(-u,-v)=\pi-\Omega(u,-v)$.

*The level sets of $\Omega$ are very nice.

*If one fixes any inner product on $V$, one can identify $S$ with a sphere $S^{n-1}\subset\mathbb R^n$ which has a natural smooth manifold structure. The metric $\Omega$ on this manifold actually arises from a Riemannian metric.


I don't know how to pack all this into nice axioms, so that any angle function satisfying them would arise from an inner product.
