Right-invariant coframe of the real Heisenberg group
= Right-invariant coframe of the real Heisenberg group
{title2=$dx,dy,dz-y\,dx$}
For the <Heisenberg group> law $(x,y,z)(a,b,c)=(x+a,y+b,z+c+xb)$, the right <Maurer-Cartan form> has coefficients $dx,dy,dz-y\,dx$. They satisfy $d\sigma^1=d\sigma^2=0$ and $d\sigma^3=\sigma^1\wedge\sigma^2$. A general <right-invariant Riemannian metric> is a constant positive-definite symmetric matrix in this coframe.