Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 25I / a / Solution

A twice differentiable curve $\gamma :I\to S$ is a geodesic when its velocity is parallel along the curve:For a surface embedded in ${\mathbb{R}}^3$, the ambient acceleration criterion for a surface geodesic gives the equivalent conditionThe parameter $t$ is then an affine parameter, and $|\dot{\gamma }|$ is constant.