In geometry, it seems that visual proofs are much more acceptable than in for example algebra (for good reason). My question is about when visual proofs are sufficient in the context of mathematics research papers. I will give an explicit example to make the question more directed:
Theorem: If we consider the unit circle, in $\mathbb{C}$, i.e. the set $$ S=\{z\in\mathbb{C} | |z| = 1\} $$ As a subspace of the metric $\mathbb{C}$, then the intrinsic metric on $S$ is given by \begin{align} d_{intr}: S^2 &\to S \\ (z_1,z_2) &\mapsto \theta \end{align} Where $\theta$ is the smallest angle between the two numbers in radians.
From a formal point of view, this is quite a bold statement, the intrinsic metric on some connected subspace is not a trivial concept, and to formally prove this theorem, would likely require a decent amount of tedious effort: First you need to prove that there is a path between any two points, and then you need to show that the infimum of all path lengths of paths connecting two particular points agrees with the given expression.
From a less formal point of view, this is just a circle, and the length of an arc is given by the angle that it stretches over, which is the angle between the two numbers. A justification like this could be accompanied for example with a sketch that shows two numbers, and colours the shortest path between them, with the relevant angle drawn in as well. This is clearly not a formal proof, but it is convincing.
My question is, in the context of a research article, is such a justification sufficient for something which is as "apparent" as the above statement, or is there the expectation that even statements like these should be accompanied by a proper proof, or at least a reference?
(I did consider posting this on academia stack exchange instead, but the specific example might not be very effective there, and it is quite relevant to my concern)