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For concreteness, let us consider the surface described in this question: it is the graph of

$$ z=\sin(x)+\sin(y) $$

The metric on the surface is

$$ g=\left( \matrix{1+\cos^2(x) & \cos(x) \cos(y) \\ \cos(x) \cos(y) & 1+\cos^2(y)} \right) $$

I am more physicist than mathematician. I'm familar with GR, but not differential geometry. I'm quite happy finding the Christoffel symbols, curvature tensor, etc. I am aware of a theorem that guarantees unique geodesics at a point and in a given direction, (modulo certain niceties) but I'm interested in uniqueness between two points.

In this answer, it seems that there exists a length minimizing geodesic, but not necessarily that it is unique.

This answer mentions the Cartan-Hadamard theorem. It seems for a surface, 'sectional curvature'=scalar curvature. For the surface in question

$$ R=\frac{8 \sin(x) \sin(y)}{(4+\cos(2x)+\cos(2y))^2} $$

Which oscillates between positive and negative over the surface. Can we apply the theorem to a region of the surface where $R<0$, and say that for two points within that region, there is a unique length minimizing geodesic?

Is there any way to tell if geodesics connecting two given points on surface are unique? Can this be phrased in terms of: "requirements for the metric such that there are unique geodesics"?

Specifically, I'm asking about 'nice' surfaces like the one here: it doesn't have points removed, and it's smooth (you will undoubtedly shake your head and tell me the correct term to use).

Sal
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    This is a really good question. I do not have an answer at the moment. When thinking about this type of thing I imagine pulling a string restricted to lie in the surface tight and I could totally see two different strings wrapping on opposite sides of those bumps and having the same length, the only question being whether they are longer or shorter than going right over the bump. I don't know if that very vague intuition captures parameters of interest here. – leslie townes Feb 20 '21 at 21:17
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    This surface, being a graph over the $xy$-plane, is simply connected. Notice that for a non-simply connected surface, e.g., a hyperboloid of one sheet, you can have two distinct geodesics joining two points (consider points on the waist of the hyperboloid). – Ted Shifrin Feb 20 '21 at 21:17
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    @leslietownes Exactly! One could imagine that on a surface like that the geodesics between 'peaks' are unique but the geodesics between 'troughs' are not, or vice versa, and I feel (optimistically?) that there should be a way to tell – Sal Feb 20 '21 at 21:30
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    @leslietownes, I agree that for the specific surface described, it's not obvious whether the geodesic from the top of one saddle to the top of another goes over the peak of the hill or not. However, it's pretty clear that if we increase the height of the peak enough, then the shortest geodesic will definitely go around instead of over the hill. In that case, there are two different shortest geodesics. – Deane Feb 20 '21 at 21:31
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    @TedShifrin I see it, but being simply connected by itself can't be enough, right? Because what about the sphere: it's simply connected but opposite points don't have unique geodesics – Sal Feb 20 '21 at 21:31
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    @Deane yeah. There's definitely symmetry here which means if the shortest path is not straight through (along a line fixed by a reflection) non-uniqueness would be guaranteed. – leslie townes Feb 20 '21 at 21:37
  • @Sal: Of course. Positive curvature is quite different from negative!! :) – Ted Shifrin Feb 20 '21 at 21:37
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    Figuring out whether there is a unique shortest geodesic joining two points is in general very hard to do. If there is enough symmetry in the surface, then sometimes that helps. In a surface where you have a hill, where for each curve going around one side of the hill, there is a congruent curve going around the other side, it suffices to show that the curve going over the top of the hill is not the shortest geodesic. This can be sometimes be done by looking at essentially the second derivative of the length function via what are known as Jacobi fields. – Deane Feb 20 '21 at 21:38
  • @TedShifrin Ah, right, of course! And the problem here is the curvature is nonconstant – Sal Feb 20 '21 at 21:59
  • @Deane Okay, I see how that would work here and in other symmetric situations. I was hoping for a more general test, but I accept that in general it might not be possible – Sal Feb 20 '21 at 22:01
  • @Sal: Constancy of curvature isn't relevant, of course. – Ted Shifrin Feb 20 '21 at 22:09
  • @TedShifrin Perhaps I should have said: "here the curvature changes sign over the surface"? – Sal Feb 20 '21 at 22:12
  • Yes, right, I was referring only to your discussion of staying in the negatively curved region. To be honest, I haven't thought about the question much. – Ted Shifrin Feb 20 '21 at 22:30
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    The problem can be very hard even for surfaces which are just diffeomorrphic to $\mathbb{R}^2$; even if you restrict to surfaces of the form $z =f(x, y) $ with $f$ smooth the uniqueness is hard. In particular, it can't be controlled by local data like curvature or stuff, because it depends on how things "collaborate" globally. Of course there could be some sufficient local criteria to establish uniqueness, like being just a hill or just a valley. I think these cases correspond somehow to convexity of the associated lagrangian. In your case, note that the the height of the hill... – Andrea Marino Feb 20 '21 at 23:32
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    ... decides uniqueness: if it is very high, it's not unique (two tours) ; if it is very small is unique (straight path) ; if it has the right height, I think you have a plethora of paths like in the sphere. As you can see, this family of surfaces have qualitatively all the same local data but the answer is different. If you put hills of different heights at any point, non-uniqueness can pop up for very large scale geodesics, but not for small scale. – Andrea Marino Feb 20 '21 at 23:35
  • @AndreaMarino Right, so it's looking like each such problem must be studied individually. Which itself is useful information. – Sal Feb 21 '21 at 12:04

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