I've looked through many similar questions on this site and also searched Google, but I couldn't find a precise definition that works for what I'm about to ask.
I'm talking about conformal maps from $\mathbb{R}^n$ to $\mathbb{R}^n$, not in the complex sense, I already know the relation between a holomorphic map and conformality, but this is not the object of the question. So, I understand the concept of conformal map at a point. It locally preserves angles between curves and their orientation. It can be defined as a differentiable map at that point whose Jacobian is a possitive multiple of a rotation matrix. And this is the best definition for now because I couldn't find another good-enough equivalent definition.
Trying to formalise the idea of "preserves angles", it would look like this:
Let $f : D\subseteq\mathbb{R}^n \to \mathbb{R}^n$ where $D$ is open. Take $\gamma : I \to D$ to be a curve through the point ${\bf x_0}$, $\gamma(t_0)={\bf x_0}$ and $\gamma$ is differentiable at ${\bf x_0}$.
The function $f$ is conformal at ${\bf x_0}$ if there exists a rotation matrix ${\bf R}$ so that for any such $\gamma$, $f\circ\gamma$ is differentiable at $t_0$ and $(f\circ\gamma)'(t_0) = \alpha{\bf R}\gamma'(x_0),\ \ \alpha\in\mathbb{R}_{>0}$
In this form, the definition with Jacobian just implies it. It's necessary, but not sufficient. Guided by several other approaches I've seen, let's restrict the function $f$ to map continuously-differentiable curves $\gamma$ at ${\bf x_0}$ to continuously-differentiable curves at $f({\bf x_0})$. This way, I think the definitions are equivalent. However, I've never seen this condition stated anywhere.
Another thing is that in some sources, the only curves considered were smooth paths. Why would we add this restriction?
Lastly, let's suppose instead that $D$ is not open, but ${\bf x_0} \in D$ is an accumulation point of $D$. Would this work?
Could anyone provide a definition that is equivalent to $f$ is differentiable at ${\bf x_0}$ and the Jacobian is a positive scalar times a rotation matrix, but regarding curves and angles?
Best Answer
Classically I've seen a conformal map defined as a differentiable map $\phi: M\to N$ for $M,N$ Riemannian manifolds with metrics, $g, \tilde{g}$ such that $$\tilde{g}(\phi_*(v),\phi_*(w))=\lambda^2(p)g(v,w)$$ for $v,w\in T_pM$, where $\phi_*$ is the pushforward of vectors under $\phi$, and $\lambda: N\to \mathbb{R}$ is differentiable.
In $\mathbb{R}^n$ to $\mathbb{R}^m$ Then we have $\phi: \mathbb{R}^n\to\mathbb{R}^m$ such that $J_{\phi}v\cdot J_{\phi} w=\lambda^2(x)(v\cdot w)$ for $v$ and $w$ tangent vectors to curves at $\mathbb{R}^m\ni x=\phi(y)$, $y\in\mathbb{R}^n$. We require differentiability because in general, it is difficult to talk about the orientation at which two curves intersect one another without requiring them to be differentiable. I don't think there is a more basic way to define such a map and remove the differentiability condition.