# Anti-symmetric Christoffel symbol

• A
• mertcan
In summary, in symmetric christoffel symbols, the torsion tensor vanishes; in anti-symmetric christoffel symbols, the torsion tensor does not vanish.

#### 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??

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.

Orodruin said:
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??

mertcan said:
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.

mertcan said:
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.

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

## 1. What is the Anti-symmetric Christoffel symbol?

The Anti-symmetric Christoffel symbol is a mathematical concept used in differential geometry to represent the connection between covariant and contravariant components of a tensor field.

## 2. Why is the Anti-symmetric Christoffel symbol important?

The Anti-symmetric Christoffel symbol plays a crucial role in the theory of general relativity, where it is used to describe the curvature of spacetime and the behavior of particles under the influence of gravity.

## 3. How is the Anti-symmetric Christoffel symbol calculated?

The Anti-symmetric Christoffel symbol is calculated by taking the difference between two symmetric Christoffel symbols, which are themselves calculated from the metric tensor and its derivatives.

## 4. Can the Anti-symmetric Christoffel symbol be visualized?

No, the Anti-symmetric Christoffel symbol cannot be visualized as it is a mathematical abstraction used to represent the connection between different components of a tensor field.

## 5. Are there any applications of the Anti-symmetric Christoffel symbol outside of physics?

Yes, the Anti-symmetric Christoffel symbol has applications in fields such as differential geometry, where it is used to study the behavior of manifolds, and in computer science, where it is used in algorithms for data analysis and machine learning.