Ambient matrix multiplication
$$
M_{2n}(\mathbb R)\times M_{2n}(\mathbb R)\longrightarrow M_{2n}(\mathbb R),
\qquad(A,B)\longmapsto AB,
$$
is 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-115/3/c/solution