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What are the different definitions of torsion-free connections in GR?

I am summarizng torsion-less condition in various context, if you know anything about it please let me know.

  1. One typical definition of torsion is \begin{align} T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y] \end{align} with this one can say torsion \begin{align} T_{MN}{}^K = 2 \Gamma_{[MN]}{}^K = \Gamma_{MN}{}^K - \Gamma_{NM}{}^K \end{align} and demand that $T_{MN}{}^K=0$, we have symmetrized $\Gamma$ for first two indices (In above notation).

    I can re-write this as \begin{align} [X,Y](\partial) = [X,Y](\nabla) \end{align} where $[X,Y]$ are usual Lie bracket.

  2. In Physics, focusing on general relativity, they consider torsion-less connection from the beginning. In some textbook, states via geodesic equation \begin{align} \Gamma_{\mu\nu}{}^\lambda = \frac{\partial x^\lambda}{\partial \xi^{\alpha}} \frac{\partial^2 \xi^\alpha}{\partial x^\mu \partial x^\nu} \end{align} or other textbook states in terms of orthonormal basis \begin{align} \nabla_{e_i} e_j = \nabla_{e_j} e_i = \Gamma_{ij}{}^k e_k \end{align} Above two connections imply torsion-less in its definition. (Of course, one can distinguish $\nabla_{e_i} e_j$ and $\nabla_{e_j} e_i$. $i.e$, One can define torsion as a subtraction between them. )

  3. Or some textbook as Carroll, defines torsion as a commutation relation from covariant derivatives $i.e$, \begin{align} [\nabla_A, \nabla_B] V_K = - R_{MNK}{}^L V_L - T_{MN}{}^K \nabla_L V_K \end{align} apparently, with some computation we see that \begin{align} T_{MN}{}^K = 2 \Gamma_{[MN]}{}^K \end{align} which agrees with first statement.

  4. Or one can deduce this from \begin{align} L_X(\partial) = L_X(\nabla) \end{align} where $L_X$ is a Lie-deriative along $X$ direction. Applying for $V_K$ we have $\Gamma_{MN}{}^K = \Gamma_{NM}{}^K$

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closed as unclear what you're asking by John Rennie, knzhou, ACuriousMind, user36790, ja72 Jul 27 '16 at 18:44

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    $\begingroup$ Would it be a good idea to include a specific question in your post? It looks like , to me anyway, you want to discuss the pros and cons of the various definitions. $\endgroup$ – user108787 Jul 27 '16 at 6:54
  • $\begingroup$ If you want to turn your findings into a resource, the most common way to do it is to post a question (like, "what are the different definitions of torsion-free connections in GR?"), answer your own question (using the stuff you posted above), and accept that answer. $\endgroup$ – knzhou Jul 27 '16 at 8:26
  • $\begingroup$ Otherwise, it's unclear what we should do with this question... $\endgroup$ – knzhou Jul 27 '16 at 8:26
  • $\begingroup$ @knzhou, Thanks! That was my intention!. I changed my question $\endgroup$ – phy_math Jul 27 '16 at 13:10
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    $\begingroup$ ...and the question here is? $\endgroup$ – ACuriousMind Jul 27 '16 at 13:37