Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 1 / b / Solution

In a coordinate chart centred at the identity, smooth multiplication has a local group lawThe identity gives $F(0,x)=x$ and $F(y,0)=y$, while associativity gives the functional equation $F(u,F(y,x))=F(F(u,y),x)$. Smooth inversion determines coordinates $\bar{x}$ with $F(\bar{x},x)=F(x,\bar{x})=0$. Changes of local coordinates alter the symmetric part of $B^k{}_{ij}$, whereas its antisymmetric part supplies the coordinate-independent Lie bracket on the tangent space at the identity.