Editing Geodesic (section)

===Existence and uniqueness===
The ''local existence and uniqueness theorem'' for geodesics states that geodesics on a smooth manifold with an [[affine connection]] exist, and are unique.  More precisely:

:For any point ''p'' in ''M'' and for any vector ''V'' in ''T<sub>p</sub>M'' (the [[tangent space]] to ''M'' at ''p'') there exists a unique geodesic <math>\gamma \,</math> : ''I'' &rarr; ''M'' such that
::<math>\gamma(0) = p \,</math> and
::<math>\dot\gamma(0) = V,</math>
:where ''I'' is a maximal [[open interval]] in '''R''' containing 0.

The proof of this theorem follows from the theory of  [[ordinary differential equation]]s, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the [[Picard&ndash;Lindelöf theorem]] for the solutions of ODEs with prescribed initial conditions. γ depends [[smooth function|smoothly]] on both ''p'' and&nbsp;''V''.

In general, ''I'' may not be all of '''R''' as for example for an open disc in '''R'''<sup>2</sup>. Any {{mvar|γ}} extends to all of {{mvar|ℝ}} if and only if {{mvar|M}} is [[geodesic manifold|geodesically complete]].