Is it mathematically correct to say that if the metric is flat/curved the shortest path is/not a Euclidean straight line?
I am still hesitant to make this claim, due to at least one counter example. One counter example was discovered in a previous post, stating that hyperbolic spaces can have straight lines who maximize distance between 2 points.
(1) Are they any other counter examples to this claim?
(2) Should I revise the statement by replacing shortest with arc-extrema?
First Edit
When I said Euclidean straight line, I meant to say that the line is straight when measured with respect to the Euclidean metric $ds^2=(dx_1^2+dx_2^2+\cdots)$. I purposely put Euclidean and straight together to avoid confusion with what straight means by itself. One could argue that the equilateral arc on a sphere is a straight line relative to the sphere's surface metric $ds^2=(d\theta^2+\sin^2(\theta)d\phi^2)$.
Also I do not think folding Euclidean space like origami counts as a counter example for shortest paths in Euclidean space, because that's equivalent to embedding the path to a constrained surface in a lower dimension which bends at district regions of space. The metric would be point-wise curved?
Second Edit
I define a metric to be flat when $R_{abcd}(g_{mn})=0$ and a curved metric to be $R_{abcd}(g_{mn})\neq 0$.
I think my problem is I am using Euclidean straight lines to encompass both Euclidean and hyperbolic, but this is not correct. It might have been better to say "straight lines in a flat metric?"
The cylinder is a good counter example. Its metric $ds^2=dz^2+R^2d\phi^2$ is flat, but its minimum paths are circular arcs? This raise another good question; are circular arcs considered the shortest path in this example? To repeal this counter example I would have to include a statement that rules out surfaces that are not minimally embedded?