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

A standard computation shows that $\displaystyle\begin{gathered} \widetilde \Gamma _{ij}^k = \frac{1}{2}{\widetilde g^{kl}}\left( {{{\widetilde g}_{il,j}} + {{\widetilde g}_{jl,i}} - {{\widetilde g}_{ij,l}}} \right) \hfill \\ \qquad= \frac{1}{2}{e^{ - 2f}}{g^{kl}}\left( {\frac{\partial }{{\partial {x^j}}}({e^{2f}}{g_{il}}) + \frac{\partial }{{\partial {x^i}}}({e^{2f}}{g_{jl}}){g_{jl,i}} - \frac{\partial }{{\partial {x^l}}}({e^{2f}}{g_{ij}})} \right) \hfill \\ \qquad= \frac{1}{2}{e^{ - 2f}}{g^{kl}}\left( {2{e^{2f}}\frac{{\partial f}}{{\partial {x^j}}}{g_{il}} + {e^{2f}}{g_{il,j}} + 2{e^{2f}}\frac{{\partial f}}{{\partial {x^i}}}{g_{jl}} + {e^{2f}}{g_{jl,i}} - 2{e^{2f}}\frac{{\partial f}}{{\partial {x^l}}}{g_{ij}} - {e^{2f}}{g_{ij,l}}} \right) \hfill \\ \qquad= \Gamma _{ij}^k + {g^{kl}}\left( {\frac{{\partial f}}{{\partial {x^j}}}{g_{il}} + \frac{{\partial f}}{{\partial {x^i}}}{g_{jl}} - \frac{{\partial f}}{{\partial {x^l}}}{g_{ij}}} \right) \hfill \\ \qquad= \Gamma _{ij}^k + \left( {\delta _i^k\frac{{\partial f}}{{\partial {x^j}}} + \delta _j^k\frac{{\partial f}}{{\partial {x^i}}} - {g_{ij}}{g^{kl}}\frac{{\partial f}}{{\partial {x^l}}}} \right). \hfill \\ \end{gathered}$

Thus, $\displaystyle\widetilde \Gamma _{ij}^k = \Gamma _{ij}^k + \left( {\delta _i^k\frac{{\partial f}}{{\partial {x^j}}} + \delta _j^k\frac{{\partial f}}{{\partial {x^i}}} - {g_{ij}}{g^{kl}}\frac{{\partial f}}{{\partial {x^l}}}} \right).$

If we use the following conformal change $\varphi^\frac{4}{n-2}$ instead of $e^{2f}$, we then see that $\displaystyle {e^f}{\partial _i}f = \frac{2}{{n - 2}}{\varphi ^{ - \frac{{n - 4}}{{n - 2}}}}{\partial _i}\varphi$

that is $\displaystyle {\partial _i}f = \frac{2}{{n - 2}}{\varphi ^{ - \frac{{n - 4}}{{n - 2}}}}{\varphi ^{ - \frac{2}{{n - 2}}}}{\partial _i}\varphi = \frac{2}{{n - 2}}{\varphi ^{ - 1}}{\partial _i}\varphi .$

In other words, $\displaystyle\widetilde \Gamma _{ij}^k = \Gamma _{ij}^k + \frac{2}{{n - 2}}{\varphi ^{ - 1}}\left( {\delta _i^k{\partial _j}\varphi + \delta _j^k{\partial _i}\varphi - {g_{ij}}{g^{kl}}{\partial _l}\varphi } \right).$

1. ..Christoffel published papers on function theory including conformal mappings geometry and tensor analysis Riemanns o-function the theory of invariants orthogonal polynomials and continued fractions differential equations and potential theory light and shock waves. ..Some of Christoffels early work was on conformal mappings of a simply connected region bounded by polygons onto a circle.

Comment by Panama corporation — May 28, 2011 @ 6:32

2. There is a typo-ish mistake in the second term in the paranthesis in the second line of the transformed Christoffel symbol.

Comment by Peter — March 29, 2020 @ 2:26

• Thank you. That term $\displaystyle\frac{\partial }{{\partial {x^i}}}({e^{2f}}{g_{jl}}){g_{jl,i}}$

should be $\displaystyle\frac{\partial }{{\partial {x^i}}}({e^{2f}}{g_{jl}}).$

Comment by Ngô Quốc Anh — March 29, 2020 @ 2:32

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