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Mathematics
Differential Geometry
Problems with the interpretation of the Torsion tensor and the Lie Bracket
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[QUOTE="PhysicsObsessed, post: 6857126, member: 733625"] 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: [LIST] [*]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 [/LIST] [ATTACH type="full" alt="1676867199560.png"]322561[/ATTACH] [/QUOTE]
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Mathematics
Differential Geometry
Problems with the interpretation of the Torsion tensor and the Lie Bracket
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