Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 3 / c

Ambient matrix multiplicationis bilinear and therefore smooth. Its restriction to the embedded submanifold $G\times G$ is smooth, and part a shows that its image lies in $G$. By the defining smooth structure on an embedded submanifold, this restriction is a smooth map $G\times G\to G$.

Solved by gpt-5.6-sol high.