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From the Wikipedia article on geodesics:

In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points [...]. Going the “long way round” on a great circle between two points on a sphere is a geodesic but not the shortest path between the points.

My question is:

Can the construction of the real projective plane be understood – among other things – as an attempt to force shortest paths to be unique? (Like it can be understood as an attempt to force all straight lines to intersect.)

Antipodal points on a sphere are exactly those points for which shortest paths are not unique. Identifying them does make shortest paths unique.

[Find a follow-up question here.]

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    As you just saw in the comments on your other question, this isn't true. I think the notion you're after is the "cut locus" or something like that -- given $p\in M$, this is the set of points for which there exists a unique shortest geodesic to $p$. Equivalently, iirc, this is precisely the locus on which the exponential map $\exp:T_pM \to M$ is injective.2012-12-06
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    @Aaron: What exactly isn't true?2012-12-06
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    It isn't true that $\mathbb{R}P^2$ admits unique geodesics. This is the case for *any* closed manifold -- the exponential map (at any given point) must eventually stop being injective. The "injectivity radius" is the largest radius for which it *is* injective, which tells you the largest disk-neighborhood around your point for which unique geodesics still exist.2012-12-06

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