Why does a geodesic $\gamma$ have unit speed and $\ddot{\gamma}$ is normal to $S$ at each point

Question

Hi i am reading about geodesics and there in the definition it is mentioned that

A curve $\gamma$ on a surface $S$ is called a geodesic if :
(a) it has unit speed
(b) $\ddot\gamma$ is normal to $S$ at each point.

My question is that is there any particular reason for choosing these conditions or are they arbitrary. Why unit speed and why is $\ddot\gamma$ is normal to $S$ at each point?

Answer 1

First the unit speed is a choice of parametrization that is made mostly for convenience. One can define different curves $c_1: [0,1] \rightarrow M$, $c_2: [0,1] \rightarrow M$ that actually define the same set of points on the manifold $M$. Unit speed fixes a particular parametrization.

Second second derivative being normal to the curve is the property that makes a geodesic a geodesic (assuming it is unit speed). This is equivalent to saying that it locally minimizes distance. For any two points sufficiently close to each other a sufficiently short curve between them is a geodesic if and only if it minimizes the distance between the points. One has to prove that this is equivalent to the normal second derivative but this is the reason why we care about geodesics.