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Can Riemann curvature tensor be non-zero in flat space

  1. if the 3rd term (Lie bracket term) is non zero,

  2. if the 3rd term is zero?

I was experimenting with random vector in flat space (Minkowoski metric, basis vectors are constant) and found that we can get non zero Riemann when the Lie Bracket term is non zero. If I am not mistaken, what does it even imply?

Edit: Here I mean Riemann in general, using the equation: $$R(u, v)w = \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{[u,v]}w $$

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  • $\begingroup$ What is the 3rd term? You should give the expression of the Riemann curvature tensor that you are referring to. $\endgroup$
    – Christophe
    Commented Nov 23, 2022 at 18:08
  • $\begingroup$ Edited and added the equation. $\endgroup$
    – Nayeem1
    Commented Nov 23, 2022 at 18:23

1 Answer 1

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The Riemann tensor is build of Christoffel symbols and their derivatives. In flat space, there is always a choice of coordinates such that Christoffel symbols are zero, so the Riemann tensor is also zero, which means that it vanishes in any coordinate system.

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