# 测地线本文重定向自 测地线

3条测地线构成的球面三角形。在球面上，测地线是大圆

## 微分几何的测地线

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

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

### 唯一性及存在性

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

${\displaystyle \gamma :[a,b]\to M}$是一条测地线，${\displaystyle -\infty 。如果对起点${\displaystyle \gamma (0)}$及起点的切向量${\displaystyle {\dot {\gamma }}(0)}$改变得足够细微，则存在新的测地线符合新的初值条件，且仍然定义在${\displaystyle [a,b]}$上。这个结果用严格语言叙述为：

## 度量几何的测地线

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

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

${\displaystyle \gamma _{1}}$是从（0,0）到（1,0）再到（1,1）的两条线段所组成，而${\displaystyle \gamma _{2}}$是从（0,0）到（2,0）的线段。这两条都是测地线，且在（0,0）到（1,0）一段重合，但明显不属同一条测地线，因为这两条线过了点（1,0）之后就分开。

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

## 参考

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.