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Hi,

I've been doing a course on Tensor calculus by Eigenchris and I've come across this problem where depending on the way I compute/expand the Lie bracket the Torsion tensor always goes to zero. If you have any suggestions please reply, I've had this problem for months and I'm desperate to solve it.

I tried computing an actual example on a spherical manifold with some simple vector fields u and v to see if that would help clarify the issue, however it didn't turn out. Though I believe the problem lies with the u(v) being equal to ∇_u(v), which shouldn't be the case.

I screenshotted the problem below, but I'll add some clarification on the notation here:

I've been doing a course on Tensor calculus by Eigenchris and I've come across this problem where depending on the way I compute/expand the Lie bracket the Torsion tensor always goes to zero. If you have any suggestions please reply, I've had this problem for months and I'm desperate to solve it.

I tried computing an actual example on a spherical manifold with some simple vector fields u and v to see if that would help clarify the issue, however it didn't turn out. Though I believe the problem lies with the u(v) being equal to ∇_u(v), which shouldn't be the case.

I screenshotted the problem below, but I'll add some clarification on the notation here:

- vector / vector field: any letter with a harpoon on top
- partial derivative operator (w respect to coordinate variables): del
- covariant derivative: nabla symbol
- connection coefficients / Christoffel symbols: capital gamma
- contravariant component: the index will be a superscript (see the vector field components u^i)
- covariant component: the index will be a subscript (see the basis vectors / partial derivatives del_i)
- Torsion tensor: T (capital T)
- Lie bracket: [] (square brackets)
- Also, I'm using index notation / Einstein notation to represent summations

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