= Curvature of a conformally Euclidean metric
{title2=$g_{ij}=e^{2u}\delta_{ij}$}
For a positive conformal factor in dimension $n\geq3$, set $S_{ij}=-u_{ij}+u_i u_j-\tfrac12|\nabla u|^2\delta_{ij}$. These are the coordinate components of the <Schouten tensor>. In the convention $R_{ijkl}=g(e_i,R(e_k,e_l)e_j)$, the <Riemann curvature tensor> is $R_{ijkl}=e^{2u}(\delta_{ik}S_{jl}+\delta_{jl}S_{ik}-\delta_{il}S_{jk}-\delta_{jk}S_{il})$. The <orthonormal coframe> $e^u dx^i$ has <connection 1 forms> $\omega^i{}_j=u_jdx^i-u_idx^j$; substituting into <Cartan's second structure equation> proves this expression directly. Its contractions give $R_{ij}=(n-2)S_{ij}+\delta_{ij}\sum_kS_{kk}$ and $R=e^{-2u}[-2(n-1)\Delta u-(n-1)(n-2)|\nabla u|^2]$.
Back to article page
