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**Need help! MTW's Gravitation exercise 16.1!**

I am working on MTW's Gravitation and I came across a problem. In the attachment https://www.physicsforums.com/attachments/50292, I show my calculation and, finally, I show the contradiction that is,

you can get a vanishing [itex]\Gamma^k_{\phantom{k}kk}[/itex] with Cartan's equation

but, at the same time,

you can also get a non-vanishing [itex]\Gamma^k_{\phantom{k}kk}[/itex] by simply inserting metric into the definition of connection coefficients.

What's wrong with my calculation? I need your help. Thank you!

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