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## Main Question or Discussion Point

Hello!

Here and there I find that it is possible to make the Christoffel symbols vanish on a curve (e.g. lecture http://www.phys.uu.nl/~thooft/lectures/genrel_2010.pdf" [Broken] by 't Hooft).

The transformation law of the Christoffel symbols is relevant in this case:

[tex]\tilde{\Gamma}_{\kappa\lambda}^{\nu}(u(x))=u_{\:,\mu}^{\nu}\left[x_{\:,\kappa}^{\alpha}x_{\:,\lambda}^{\beta}\Gamma_{\alpha\beta}^{\mu}(x)+x_{\:,\kappa,\lambda}^{\mu}\right].[/tex]

We can fine-tune the second term in the square parentheses to cancel the first term and then [tex]\tilde{\Gamma}[/tex] vanishes. But how does one see it is only possible to do that only on a curve and not the whole space? What about more (or less) than four dimensions?

Thanks!

Here and there I find that it is possible to make the Christoffel symbols vanish on a curve (e.g. lecture http://www.phys.uu.nl/~thooft/lectures/genrel_2010.pdf" [Broken] by 't Hooft).

The transformation law of the Christoffel symbols is relevant in this case:

[tex]\tilde{\Gamma}_{\kappa\lambda}^{\nu}(u(x))=u_{\:,\mu}^{\nu}\left[x_{\:,\kappa}^{\alpha}x_{\:,\lambda}^{\beta}\Gamma_{\alpha\beta}^{\mu}(x)+x_{\:,\kappa,\lambda}^{\mu}\right].[/tex]

We can fine-tune the second term in the square parentheses to cancel the first term and then [tex]\tilde{\Gamma}[/tex] vanishes. But how does one see it is only possible to do that only on a curve and not the whole space? What about more (or less) than four dimensions?

Thanks!

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