# Ngô Quốc Anh

## May 22, 2011

### Conformal changes of Christoffel symbols

Filed under: Riemannian geometry — Ngô Quốc Anh @ 18:10

In this short note, we try to calculate conformal changes of Christoffel symbols that we have touched in the previous note [here]. Let us pick two Riemannian metrics $g$ and $\widetilde g$ on a manifold $M$ sitting in the same conformal class, that is,

$\displaystyle \widetilde g=e^{2f}g$

where $f$ is a smooth function on $M$.

Let us recall that Christoffel symbols are determined by

$\displaystyle \Gamma _{ij}^k = \frac{1}{2}{g^{kl}}\left( {{g_{il,j}} + {g_{jl,i}} - {g_{ij,l}}} \right)$

where

$\displaystyle {g_{,m}} = \frac{{\partial g}}{{\partial {x^m}}}.$