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  1. The symmetry of basis vectors leads to the property that the Christoffel symbols are symmetric with respect to the bottom indices (Ref: Mathematical methods for Physicists by Arfken, Weber and Harris). (Γij = Γji). The symmetry of basis vectors is a property based on geometry of space.

    (A flat space supports this geometrical property.)

    This property is used in the derivation of the Christoffel symbols formula. The formula therefore becomes dependent upon geometry of space.

  2. Therefore, is it possible to get a non-zero value for the Torsion tensor, (Tij = Γji - Γij)?

  3. Is it possible to derive a Christoffel symbol formula which is not symmetric, so that the Torsion tensor is non-zero? What will be the associated coordinate system and geometrical picture?

Question:

What I wish to know is the "Geometrical Picture" behind a non-zero torsion tensor. What will be the non-symmtric Christoffel symbol formula to obtain it? How this formula can be derived from geometry of space?

Torsion Tensor calculationsPlease see the attachment.

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    $\begingroup$ Does this answer your question? Torsion tensor in Relativity physics.stackexchange.com/q/212298 $\endgroup$
    – Eletie
    Commented Apr 23, 2021 at 7:54
  • $\begingroup$ Question has been edited to clarify the querry. $\endgroup$
    – NSRG
    Commented Apr 23, 2021 at 8:37
  • $\begingroup$ There are quite a lot of questions on here about non Levi-Civita connections & geometric interpretations of torsion instead of curvature $\endgroup$
    – Eletie
    Commented Apr 23, 2021 at 8:56
  • $\begingroup$ This isn't what we do - in GR in general we assume a Levi-Civita connection (zero torsion), before even discussing any solutions. Your question & comments seem a bit confused? $\endgroup$
    – Eletie
    Commented Apr 23, 2021 at 9:08
  • $\begingroup$ In Schwarzschild solution (for space outside the mass), we assume torsion is zero. But even for this curvature, it is difficult to define a coordinate system as expressions for unit vectors can not be written. We continue to use the standard Christoffel symbols formula even though there is no geometrical support for path independence of incremental vector. Now we have a situation where torsion is non-zero. I wish to know, if we can define a coordinate system where torsion is non-zero. How to derive an appropriate Christoffel symbols formula under such situation? $\endgroup$
    – NSRG
    Commented Apr 23, 2021 at 9:10

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