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I've been going through Bernard Schutz's A First Course In General Relativity, On Chapter 5 questions atm.

Should the Christoffel Symbols for a coordinate system (say polar) be the same for vectors and one-forms in that coordinate system?

I would have thought yes, but If you calculate the Christoffel Symbols using the Basis vectors of basis one forms then you get different Christoffel symbols?

I am asking because when you do the Covariant Derivative of a vector or a one form are the Christoffel Symbols the same of different?

For Vectors i.e.

[itex]\vec{V} = v^{\alpha}\vec{e}_{\alpha}[/itex]

We can calculate the Christoffel symbols using

[itex]\frac{∂ \vec{e}_{\alpha}}{∂x^{\beta}} = \Gamma^{\mu}_{\alpha \beta} \vec{e}_{\mu}[/itex]

Where the Basis vectors for polar coordinates are

[itex]\vec{e}_{r} = Cos(\theta)\vec{e}_{x} + Sin(\theta)\vec{e}_{y}[/itex]

[itex]\vec{e}_{\theta} = -r Sin(\theta)\vec{e}_{x} + r Cos(\theta)\vec{e}_{y}[/itex]

For One-Forms i.e.

[itex]\tilde{P} = p_{\alpha}\tilde{e}^{\alpha}[/itex]

We can calculate the Christoffel symbols using

[itex]\frac{∂ \tilde{e}^{\alpha}}{∂x^{\beta}} = \Gamma^{\alpha}_{\beta \mu} \tilde{e}^{\mu}[/itex]

Where the Basis vectors for polar coordinates are

[itex]\tilde{e}^{r} = Cos(\theta)\tilde{e}^{x} + Sin(\theta)\tilde{e}^{y}[/itex]

[itex]\tilde{e}^{\theta} = - \frac{Sin(\theta)}{r}\tilde{e}^{x} + \frac{Cos(\theta)}{r}\tilde{e}^{y}[/itex]

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# Christoffel Symbols of Vectors and One-Forms in say Polar Coordinates

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