Minimizing geodesic: Difference between revisions

Latest revision as of 19:49, 18 May 2008

A metric space $M$ equipped with a metric $d$ .

A path $\gamma :[0,1]\to M$ is termed a minimizing geodesic if it is the shortest path from $\gamma (0)$ to $\gamma (1)$ .

By shortest, we mean path of minimum length where the length of a path is defined as:

$\lim \sup _{0=t_{0}\leq t_{1}\leq t_{2}\leq \ldots \leq t_{n}=1}\sum _{i}d(\gamma (t_{i}),\gamma (t_{i+1})$

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 A [[metric space]] <math>M</math> equipped with a metric <math>d</math>.
-==Definition part===
+==Definition part==
 A path <math>\gamma: [0,1] \to M</math> is termed a '''minimizing geodesic''' if it is the shortest path from <math>\gamma(0)</math> to <math>\gamma(1)</math>.