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?
Your claim is too general, not to mention that you should clarify what you mean by "Euclidean straight line".
As Joseph O'Rourke showed in his answer, there are spaces which admit a locally flat metric but do not contain any line.
On the other hand, there are spaces which can be embedded in a Euclidean space where some geodesics are straight lines (in the sense that they are straight lines in the ambient space), while some are not. For example consider an infinite cylinder or a one-sheeted hyperboloid.