Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 4 / b

Direct differentiation gives, for example,Differentiating ${\sigma }^1,{\sigma }^2$ similarly and substituting the definitions yields $d{\sigma }^1={\sigma }^2\wedge {\sigma }^3$ and $d{\sigma }^2={\sigma }^3\wedge {\sigma }^1$. Hence these Maurer-Cartan forms on the three-sphere obey