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.

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    $\begingroup$ You might be interested in Birkhoff and von Neumann's "The Logic of Quantum Mechanics". They show in Section 14 that an inner product can be recovered up to a scalar factor from an orthogonal complement operation $U\mapsto U^\perp$ satisfying some axioms. There's a restriction to dimensions $>3$ though (and a citation to an earlier result of Brauer that might be easier). $\endgroup$ – Dap Mar 12 '18 at 12:11
  • $\begingroup$ That sounds helpful: thanks! (I also first came across these issues doing some work on Weyl, so there could be interesting historical connections going on here.) $\endgroup$ – anygivenpoint Mar 13 '18 at 10:12

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.


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