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!

  • $\begingroup$ 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

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.


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 $I$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.