2010-26.md

course	course_year	question_number	tags	title	year
Geometry	IB	26	IB 2010 Geometry	Paper 2, Section II, F	2010

Suppose that $a>0$ and that $S \subset \mathbb{R}^{3}$ is the half-cone defined by $z^{2}=a\left(x^{2}+y^{2}\right)$, $z>0$. By using an explicit smooth parametrization of $S$, calculate the curvature of $S$.

Describe the geodesics on $S$. Show that for $a=3$, no geodesic intersects itself, while for $a>3$ some geodesic does so.