Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 26I / c / ii / Solution

No. The set is closed and bounded in the finite-dimensional space $S_3(\mathbb{R})$, hence compact and nonempty. A global parametrization would be a homeomorphism from a nonempty open subset of ${\mathbb{R}}^2$ onto this compact surface, but no nonempty open subset of ${\mathbb{R}}^2$ is compact.