So I'm working with differential geometry. So my book claim that
"any geodesic has constant speed".
And the proof is left as an exercise and I found the exercise in the book.
Exercise: "Prove that any geodesic has constant speed and so a very simple unit-speed reparametrization."
I know the definition of geodesic, but I don't know how to work it out.
Thanks in advance
Edit: The definition,
"Let $\gamma$ be a curve on a surface $\textbf{S}$. Then the curve is called a $\textit{geodesic}$ if $\ddot{\gamma}$ is zero or orthogonal to the tangent plane of the surface at the point $\gamma$."
Best Answer
Since geodesics have acceleration normal to the surface, the acceleration is also normal to the velocity (which is tangent to the surface). So, if our curve is $\alpha,$ then $\alpha'\cdot\alpha''=0.$ Hence, $$0=2\alpha'\cdot\alpha''=\frac{d}{dt}\left(\|\alpha'\|^2\right).$$