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I am interested in the definition of geodesics in metric spaces. A definition which seems reasonable to me is that a geodesic should locally be a distance minimizer.

Wikipedia (http://en.wikipedia.org/wiki/Geodesic#Metric_geometry) states that this naturally leads one to define a geodesic as a curve $\gamma: I \to M$, with $I\subset \mathbb{R}$ an interval and $M$ a metric space, such that for any $t \in I$ there is a neighbourhood $J$ of $t$ so that for any $t_1, t_2 \in J$ we have $$ d(\gamma(t_1), \gamma(t_2)) = v |t_1 -t_2|, $$ where $v \geq 0$ is any constant.

Am I missing something obvious? Why is it that a curve satisfying this formula is locally minimizing the distance between its points? I have no reason to think this is false, but also no intuition as to why this is true.

Thank you in advance!

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  • $\begingroup$ I see geodesics more as "straightest possible paths" on a manifold, but this would require a notion of differentiability and thus is not applicable as a concept on general metric spaces. Your idea could be a good one. A possible problem would be that you lose all notions of uniqueness. $\endgroup$ – Daniel Robert-Nicoud Dec 20 '13 at 23:42
  • $\begingroup$ You copied the statement a bit scrambled (and that changed the meaning) from Wikipedia. It should be: There is a $v>0$ such that for every $t\in I$ there is a neighborhood $J$ of $t$ such that that inequality is satisfied for every $t_1,t_2$ in $J$. $\endgroup$ – OR. Dec 20 '13 at 23:55
  • $\begingroup$ I was just pointing out that what you wrote in the question is not what Wikipedia is saying. On the other hand, I think that Wikipedia might be giving (without saying it) a characterization of rectifiable, not of minimal length. (To define geodesic in a metric space as rectifiable curve of minimal length joining two points). Let me check. $\endgroup$ – OR. Dec 21 '13 at 0:32
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Wikipedia is being a bit confusing here. Without the constant $v$, it should be obvious that the definition you gave is a good definition of a geodesic (a curve for which arc length is locally the same as distance).

The role of the constant $v$ is to allow geodesics whose length is different from the length of the domain interval $I$. Note that $v$ must be a constant which does not depend on the neighborhood $J$, since the values of $v$ must agree on overlapping neighborhoods.

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It's worth noting that the constant $v$ necessarily depends on the points $x,y\in M$ that you wish to connect with a geodesic, since the infimum of the lengths of (rectifiable) curves from $x$ to $y$ is not necessarily the same as $d(x,y)$, and may vary from $d(x,y)$ by a different factor for different choices of $x$ and $y$. With the two points chosen, I think Wikipedia's definition is pretty intuitive: a geodesic is a path that is (proportional to) an isometry of $I$ "near" each point in $I$.

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