Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 11E (Geometry)

A regularly parametrized curve $\gamma$ on an embedded surface $\Sigma$ is a geodesic when its covariant acceleration vanishes. Equivalently, in the ambient Euclidean space,for an affine geodesic parameter. A complete geodesic is one whose maximal affine parameter interval is all of $\mathbb{R}$.

Tangency along $\gamma$ meansfor every $t$. Consequently $\ddot{\gamma }(t)$ is perpendicular to the first tangent plane exactly when it is perpendicular to the second. The ambient characterization of a geodesic therefore proves the tangency invariance of an ambient-surface geodesic:

For fixed $u$, varying $v$ traces a unit circle in the radial-vertical plane centred atThus ${\Sigma }_1$ is a unit tube whose centre winds helically around the $z$-axis. Sincethe given curve is its inner helix.

Take ${\Sigma }_2$ to be the circular cylinder $x^2+y^2=1$. Along $v=\pi$,so their span is the tangent plane of that cylinder. The two surfaces are therefore tangent along $\gamma$. Moreover,which is normal to the cylinder, so $\gamma$ is a geodesic of ${\Sigma }_2$. Part (ii) now proves that it is a geodesic of ${\Sigma }_1$. This is the cylindrical-helix tangency construction.

SetAlong $v=0$,whose span is the tangent plane of the cylinder $x^2+y^2=9$. Alsois normal to that cylinder. The same tangency argument proves that $\Gamma$ is a geodesic of ${\Sigma }_1$. It is defined for every $t\in \mathbb{R}$, hence complete, and it is disjoint from $\gamma$ because their distances from the $z$-axis are respectively $3$ and $1$.