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In the complex plane there exists a very large set of smooth conformal maps, (The famous Riemann Mapping Theorem states that between any two simply connected open sets in $\mathbb{C}$ there exists a biholomorphic mapping which is also conformal).

The situation is much more bleak as soon you go up to 3 dimensions and higher where the only smooth conformal maps are the mobius transformations. This is the famous Liouville's Theorem

I was thinking about trying to find a generalization of the concept of an "angle" so that the "set of all generalized-angle preserving maps" in $\mathbb{R}^3$ would have an interesting structure larger than merely mobius transformations.

There is the obvious generalization, solid angles. In 3 dimensions these are defined for collections of 4 points at a time (we select 1 anchor point and consider the 3 vectors formed by the anchor to the 3 other non collinear points. And then take the area of the spherical triangle spanned by these 3 vectors). The solid angle can be calculated explicitly using a formula given here.

So that leads to our question.... is the set of all smooth solid-angle-conformal functions from $\mathbb{R}^3 \rightarrow \mathbb{R}^3$ a larger set than the set of mobius transformations? And does there exist a generalization of the Riemann Mapping Theorem for them?

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    $\begingroup$ Such a great question! I've vaguely wondered this myself, but wasn't sure how to articulate it. $\endgroup$ Oct 29, 2021 at 4:59

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The basic question here is a linear algebra question: which orientation-preserving linear maps $T:\mathbb{R}^3\to\mathbb{R}^3$ preserve solid angles formed by triples of vectors? (You are then asking about smooth maps whose derivative at every point is such a linear map.) Let $G\subset GL_3(\mathbb{R})$ be the group of such maps, and let $H=\mathbb{R}^+\times SO(3)\subset GL_3(\mathbb{R})$ be the group of conformal linear maps (i.e., compositions of rotations and scalings). It is clear that $H\subseteq G$; I claim that in fact $G=H$, so your maps are just the same as conformal maps.

Suppose $T\in G$; we wish to show that $T\in H$. Let $e_1,e_2,e_3$ be the standard basis vectors for $\mathbb{R}^3$. Taking a singular value decomposition of $T$, we may compose $T$ with rotations (which does not change whether $T\in H$) to assume that $T$ is diagonal. Since $T$ is orientation-preserving, we can compose it with a further rotation to assume the diagonal entries are positive, and we can compose with a scaling to assume that $T(e_1)=e_1$. We have $T(e_2)=be_2$ and $T(e_3)=ce_3$ for positive scalars $b$ and $c$; I claim that $b=c=1$ so $T$ is just the identity.

To prove this consider the solid angle formed by the unit vectors $e_1,\frac{e_1+e_2}{\sqrt{2}},$ and $e_3$. $T$ maps these vectors to $e_1,\frac{e_1+be_2}{\sqrt{2}},$ and $ce_3$, which form the same solid angle as the unit vectors $e_1,\frac{e_1+be_2}{\sqrt{1+b^2}},$ and $e_3$. But now observe that as $b$ grows from $0$ to $\infty$, the spherical triangle formed by these vectors grows strictly (we are taking the spherical triangle formed by $e_1,e_2,$ and $e_3$ and replacing the $e_2$ vertex with a point on the edge between $e_1$ and $e_2$). So, the only way the area of this triangle would be the same as the area of the triangle formed by $e_1,\frac{e_1+e_2}{\sqrt{2}},$ and $e_3$ is if $b=1$. Swapping the roles of $e_2$ and $e_3$, we similarly conclude that $c=1$.

Similar arguments show that for any $n\geq m\geq 2$, the orientation-preserving linear maps $T:\mathbb{R}^n\to\mathbb{R}^n$ that preserve "$m$-dimensional solid angles" formed by $m$ vectors are the same as the conformal linear maps.

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  • $\begingroup$ This argument makes sense in that locally all solid-angle preserving maps must look locally like conformal maps (a shift + rotation), but even in the case of $\mathbb{C}$ that is true: after all all biholomorphic functions locally look like shifts+rotations (see taylor series $f(z) = a_0 + a_1z + o(z^2)$ and similarly all mobius transformations are locally shifts+rotations. So just because the set of supported rotations is identical to mobius transformations, doesn't mean THEY ARE mobius transformations $\endgroup$ Jan 14, 2021 at 5:05
  • $\begingroup$ Or at least its not obvious to me $\endgroup$ Jan 14, 2021 at 5:06
  • $\begingroup$ Oh wait im silly, we already take as an assumption that if a map is everywhere locally belonging to that same set of shifts + rotations then it MUST be mobius. That's Liouville's theorem. So by showing these are the same, then we conclude Louiville's theorem applies to this set too. $\endgroup$ Jan 14, 2021 at 5:07
  • $\begingroup$ Nice work here! :) $\endgroup$ Jan 14, 2021 at 5:08

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