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The torsion tensor in 4 dimensions $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-S_b\delta^c_a)$ (4 components) and an axial-vector part $\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}=g^{cd}S_{[abd]}$ (4 components) and a traceless part $\hphantom0^TS_{ab}^{\hphantom0\hphantom0c}=S_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^AS_{ab}^{\hphantom0\hphantom0 c}-\hphantom0^VS_{ab}^{\hphantom0\hphantom0 c}$ (16 components).

Torsion is usually interpreted physically as a twist in space because it causes a parallelogram to fail to close. How does the vector components particularly contribute to that twist geometrically? and what happen if it is a complex quantity i.e. $S_a=S_b=i\phi(t)$?

Eris
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    Related: What is torsion in differential geometry intuitively? – YCor Sep 10 '20 at 14:10
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    (I fixed "axialvector part" to "axial vector part" but maybe it should be "axial part", please check.) – YCor Sep 10 '20 at 14:11
  • 24 components only in certain spatial dimensions, you should probably specify the dimension (I think you mean 4 dimensions, since the "axial vector part" is determined by a three form.) Also; please explain what you mean by "complex". Are you working with an almost complex manifold? – Willie Wong Sep 10 '20 at 14:32
  • Thanks YCor. Both Axial-vector and Axial are used in literature. – Eris Sep 10 '20 at 14:35
  • @WillieWong sure, I edited the question. I'm not specifically on a complex manifold but I'm trying to understand what kind of manifold would result from such a choice of a complex torsion. – Eris Sep 10 '20 at 14:37
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    How exactly are you defining torsion then? On a real differential manifold with an affine connection, the torsion tensor is manifestly real by definition. You can't just slot in complex values and hope it would work out. – Willie Wong Sep 10 '20 at 14:47

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