On the Wikipedia page for Geodesic, it's stated that a curve $\gamma : I → M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v \geq 0$ such that for any $t \in I$ there is a neighborhood $J$ of $t$ in $I$ such that for any $t_1, t_2 \in J$ we have $d(\gamma (t_1),\gamma (t_2)) = v\left|t_1-t_2\right|.$
I am wondering how any such curve $\gamma$ is locally a minimal distance function, i.e. if $x,y \in M$ is connected by a curve $\tilde{\gamma}(t) : [a,b] \to M$ and $\gamma(t) : [a,b] \to M$ is a geodesic connecting the two points, then the length of $\tilde{\gamma}$ is greater than or equal to the length of $\gamma$ with equality possible only in the case where $\tilde{\gamma}$ is a geodesic according to the definition just given.
In general, what is the intuition behind the definition of a geodesic via the statement $d(\gamma (t_1),\gamma (t_2)) = v\left|t_1-t_2\right|$?
I realize that there are geodesics which are not the distance minimizing curves, e.g. two portions of great circles in the Riemann sphere. If a surface does not have any holes (i.e. genus is zero) and is complete, does that guarantee that geodesics minimize distance globally?
Best Answer
The answer resides in the fact, which motivates the aforementioned definition of geodesic in a metric space, that for any $t_1,t_2,t_3\in J$ with $t_l<t_2<t_3$, in the situation of the first paragraph, we have
$$d(\gamma(t_1),\gamma(t_3))=v(t_3-t_1)=v(t_2-t_1)+v(t_3-t_2)=d(\gamma(t_1),\gamma(t_2))+d(\gamma(t_2),\gamma(t_3)).$$
Put into words, this means that, locally, equality holds for the triangle inequality involving points of a geodesic $\gamma$.
Let's fulfill the details. We would like to check that a geodesic $\gamma$ is locally minimizing. First, recall that the length of a curve $\gamma:[a,b]\to M$ in a metric space $M$ is defined as the supremum
$$\ell(\gamma):=\sup_P\sum_{\substack{t,s\in P\\t<s}}d(\gamma(t),\gamma(s)),$$
where $P$ runs over all partitions of $[a,b]$, i.e. all finite subsets $P\subset [a,b]$ such that $a,b\in P$.
A curve $\gamma:[a,b]\to M$ is locally minimizing if, for all $t\in [a,b]$, there's a neighbourhood $J\subset [a,b]$ of $t$ such that every curve $\tilde\gamma:[a',b']\to M$ with $[a',b']\subset J$, $\tilde\gamma(a')=\gamma(a')$, $\tilde\gamma(b')=\gamma(b')$ satisfies the inequality
$$\ell(\gamma')\le\ell(\tilde\gamma),$$
where $\gamma':[a',b']\to M$ denotes the restriction of $\gamma$ to $[a',b']$.
Now, let $\gamma:[a,b]\to M$ be a geodesic, pick some $t\in[a,b]$ and let $J\subset[a,b]$ be a neighbourhood of $t$ such that for any $t_1,t_2\in J$, $t_1<t_2$, we have
$$d(\gamma(t_1),\gamma(t_2))=v(t_2-t_1)$$
for some constant $v\ge0$.
Take $[a',b']\subset J$ and consider the restriction $\gamma':[a',b']\to M$ of $\gamma$ to $[a',b']$. Then $\ell(\gamma')=d(\gamma'(a'),\gamma'(b'))$. Indeed, for any partition $P\subset[a',b']$ we have
$$\sum_{\substack{t,s\in P\\t<s}}d(\gamma'(t),\gamma'(s))=\sum_{\substack{t,s\in P\\t<s}}v(s-t)=v(b'-a')=d(\gamma'(a'),\gamma'(b')).$$
Thus,
$$\ell(\gamma')=\sup_P\sum_{\substack{t,s\in P\\t<s}}d(\gamma'(t),\gamma'(s))=d(\gamma'(a'),\gamma'(b')).$$
Finally, take a curve $\tilde\gamma:[a',b']\to M$ such that $\tilde\gamma(a')=\gamma'(a')$, $\tilde\gamma(b')=\gamma'(b')$. Then $\ell(\gamma')\le\ell(\tilde\gamma)$. Indeed, by the triangle inequality, for any partition $P\subset[a',b']$, we have
$$d(\tilde\gamma(a'),\tilde\gamma(b'))\le\sum_{\substack{t,s\in P\\t<s}}d(\tilde\gamma(t),\tilde\gamma(s)).$$
But
$$\ell(\gamma')=d(\gamma'(a'),\gamma'(b'))=d(\tilde\gamma(a'),\tilde\gamma(b'))$$
and
$$\sum_{\substack{t,s\in P\\t<s}}d(\tilde\gamma(t),\tilde\gamma(s))\le\sup_P\sum_{\substack{t,s\in P\\t<s}}d(\tilde\gamma(t),\tilde\gamma(s))=\ell(\tilde\gamma).$$