# A Anti-symmetric Christoffel symbol

1. May 22, 2016

### mertcan

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??

2. May 22, 2016

### Orodruin

Staff Emeritus
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.

3. May 22, 2016

### mertcan

I could not entirely conceive of the example you mentioned, could you please give some mathematical notations??

4. May 22, 2016

### Markus Hanke

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.

5. May 22, 2016

### Orodruin

Staff Emeritus
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.

6. May 22, 2016

### haushofer

And a bit more advanced example: in supergravity theories the torsion does not vanish in general because of the gravitino contribution.

7. May 23, 2016

### stevendaryl

Staff Emeritus
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