测地线

微分幾何的測地線

${\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}$

${\displaystyle {\frac {d^{2}\gamma ^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {d\gamma ^{\mu }}{dt}}{\frac {d\gamma ^{\nu }}{dt}}=0}$

唯一性及存在性

${\displaystyle \gamma :(-\epsilon ,\epsilon )\to M}$

${\displaystyle \gamma :[a,b]\to M}$是一條測地線，${\displaystyle -\infty 。如果對起點${\displaystyle \gamma (0)}$及起點的切向量${\displaystyle {\dot {\gamma }}(0)}$改變得足夠細微，則存在新的測地線符合新的初值條件，且仍然定義在${\displaystyle [a,b]}$上。這個結果用嚴格語言敘述為：

度量幾何的測地線

${\displaystyle d(\gamma (t_{1}),\gamma (t_{2}))=\left|t_{1}-t_{2}\right|}$

${\displaystyle d((x_{1},y_{1}),(x_{2},y_{2}))=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|}$

${\displaystyle \gamma _{1}}$是從（0,0）到（1,0）再到（1,1）的兩條線段所組成，而${\displaystyle \gamma _{2}}$是從（0,0）到（2,0）的線段。這兩條都是測地線，且在（0,0）到（1,0）一段重合，但明顯不屬同一條測地線，因為這兩條線過了點（1,0）之後就分開。

${\displaystyle \gamma _{t_{0}}(t)={\begin{cases}(t,0)&t\leq t_{0}\\(t_{0},t-t_{0})&t_{0}\leq t\leq t_{0}+1\\(t-1,1)&t_{0}+1\leq t\leq 3\end{cases}}}$

參考

1. ^ Petersen, Peter (2006), Riemannian geometry, Graduate Texts in Mathematics, 171 (2nd ed.), Berlin, New York.
2. ^ Burago, Dmitri; Yuri Burago, and Sergei Ivanov (2001), A Course in Metric Geometry, American Mathematical Society.