Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 26F / b / Solution

Since $z>0$, the cone has the parametrizationIts coordinate derivatives satisfyso its first fundamental form is

On a sufficiently narrow angular patch, defineand map the cone to the plane point with polar coordinates $(R,\Theta )$. The Euclidean metric pulls back asThe map is therefore a local isometry from a circular cone to the plane, proving that $\Sigma$ is locally isometric to the Euclidean plane.