How to prove that the unit circle in the $x$-$y$ plane is a geodesic on the hyperboloid $x^2+y^2-z^2=1$?

Question

Is there any way (besides a graph) to prove that the unit circle in the $x$-$y$ plane is a geodesic on the hyperboloid $x^2+y^2-z^2=1$?

Answer 1

This follows from the uniqueness of geodesics and symmetry: Suppose $S \subset \Bbb R^3$ is a surface (for concreteness---there is also an abstract version of this statement) and that $S$ is symmetric across a plane $\Pi$ such that $S \cap \Pi$ is a smooth curve. Then, that curve is an (unparameterized) geodesic. There are some details to fill in, but the idea is this: Consider a geodesic $\gamma$ tangent to any vector tangent to $S \cap \Pi$. By symmetry the reflection of $\gamma$ must also be a geodesic tangent to that same vector, so by uniqueness of geodesics with given initial condition, $\gamma$ coincides with it reflection, and hence is contained in $S \cap \Pi$.