Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 11F (Geometry)

A topological surface is a Hausdorff second-countable space locally homeomorphic to ${\mathbb{R}}^2$. Pairing opposite sides of the hexagon gives neighborhoods modelled on discs, including at edge and vertex classes; compactness follows from the compact polygon quotient. Cutting and rearranging the polygon gives the usual opposite-side square, hence a torus.

The full double cone is not a surface at the origin: deleting the vertex from a small neighborhood leaves two components, unlike a punctured disc. The upper cone with $z>0$ excludes that vertex and has ordinary annular charts, so it is a surface.

Writing the upper cone as $(r\cos \theta ,r\sin \theta ,r)$ gives metricOn any polar branch, the mappulls this back to $d{s}^2+s^{2}d{\phi }^2$, so it is a local isometry. Strictly, because $\sqrt{2}\phi$ is not $2\pi$-periodic, this formula is globally single-valued on the universal polar cover and locally on ${\mathbb{R}}^2\setminus \{0\}$; that is the developing map used here.

An isometry fixing $p$ lifts locally to a Euclidean isometry fixing a lift of $p$. Compatibility with the cone's angular identification leaves only the identity and reflection across a radial line. Descending to the cone, these are the identity and reflection in the plane through the cone axis and $p$.