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}}}.

(more…)

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