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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?

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    $\begingroup$ Speaking of "a Euclidean straight line" already presumes you're considering a subset of a Euclidean space. (The example mentioned in your earlier post is not a subset of a Euclidean space, but of a plane with a Lorentz-signature metric. Incidentally, "hyperbolic space" often connotes another geometric structure.) Your claim is certainly false as stated, even for Riemannian metrics, as Joseph O'Rourke's example shows. But the deeper point is, you need to be specific about what (class of) metric you're considering before making assertions about paths and length extremization. :) $\endgroup$
    – Andrew D. Hwang
    Jun 12, 2015 at 13:50
  • $\begingroup$ @AndrewD.Hwang I made some revisions. $\endgroup$
    – linuxfreebird
    Jun 12, 2015 at 14:37
  • $\begingroup$ Check en.wikipedia.org/wiki/Geodesic $\endgroup$
    – user65203
    Jun 12, 2015 at 14:37
  • $\begingroup$ @linuxfreebird: I'm still unsure of your intent after the edit. Particularly, 1. You write "flat" and "curved", but reject some flat metrics (cylinder, folded cube). 2. You speak of "Euclidean" lines in a non-Euclidean setting. (The Lorentz metric of your other question is not Euclidean.) 3. The Lorentz plane is not curved, so the dichotomy in your first sentence seems not to match the examples you already know. Again, it would help to know exactly what spaces (and metrics) you're trying to encompass in your claim. It would also help to know how you define "flat". $\endgroup$
    – Andrew D. Hwang
    Jun 12, 2015 at 15:42
  • $\begingroup$ @AndrewD.Hwang I provided more explanation. $\endgroup$
    – linuxfreebird
    Jun 12, 2015 at 17:16

2 Answers 2

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The figure below, taken from an earlier MSE answer, shows a (green) shortest path on a locally flat metric. Is it "a Euclidean straight line"? Depends on what that phrase means...

     Shortest path unfolding
Only after unfolding the surface to a plane is it a straight segment in $\mathbb{R}^3$.

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  • $\begingroup$ I made some revisions. $\endgroup$
    – linuxfreebird
    Jun 12, 2015 at 14:37
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    $\begingroup$ @linuxfreebird: Thanks for clarifying what you meant by a "Euclidean straight line." $\endgroup$
    – Joseph O'Rourke
    Jun 12, 2015 at 15:15
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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.

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    $\begingroup$ I made some revisions. $\endgroup$
    – linuxfreebird
    Jun 12, 2015 at 14:37

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