Anti-symmetric Christoffel symbol

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Main Question or Discussion Point

hi, I have seen some examples related to christoffel symbol when it was symmetric, but I have not seen any anti symmetric christoffel symbol examples. For instance, in torsion tensor, if we have anti symmetric christoffel symbol, torsion tensor does not vanish. To sum up, in what kind of situations is christoffel symbol anti symmetric and in what kind of situations does torsion tensor remain?? Could you please give some examples??

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Orodruin
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Consider the two-dimensional sphere with the poles removed. If you define your affine connection in such a way that compass directions are preserved under parallel transport you will have a connection which is not torsion free.

Consider the two-dimensional sphere with the poles removed. If you define your affine connection in such a way that compass directions are preserved under parallel transport you will have a connection which is not torsion free.
I could not entirely conceive of the example you mentioned, could you please give some mathematical notations??

hi, I have seen some examples related to christoffel symbol when it was symmetric, but I have not seen any anti symmetric christoffel symbol examples. For instance, in torsion tensor, if we have anti symmetric christoffel symbol, torsion tensor does not vanish. To sum up, in what kind of situations is christoffel symbol anti symmetric and in what kind of situations does torsion tensor remain?? Could you please give some examples??
I am not sure about the anti-symmetry bit, but so far as torsion is concerned, the torsion tensor will in general be non-zero if you choose a connection on your manifold which is not Levi-Civitá.

Orodruin
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I could not entirely conceive of the example you mentioned, could you please give some mathematical notations??
Define the vector fields $\partial_\theta$ and $(1/\sin\theta)\partial_\varphi$ as parallel (this is sufficient to deduce the connection coefficients). The corresponding affine connection is not torsion free and corresponds to the situation described in my previous post.

It should be noted that the affine connection defind this way is metric compatible with the standard metric.

haushofer