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Suppose we have a metric tensor $g_{\mu \nu}$ defining a Riemannian manifold. The Christoffel symbols related to this metric are not tensors. Their difference anyway is a tensor: $$T^k_{ij}=\Gamma^k_{ij}-\Gamma^k_{ji}$$ This $T^k_{ij}$ defines the 'Torsion' of the connection. Why is there this link between this tensor and the torsion? Thanks in advance

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  • $\begingroup$ $T^k_{ij}$ measures how much the metric $g_{lk}$ is an-holonomic, i.e. the tangent basis is far from a coordinated one $\endgroup$ – janmarqz Nov 1 '18 at 18:05
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In advance: I know this should be a comment, but I do not have the required repution,yet...

Idk what you are asking for.

It is pretty much how torsion is defined in this setting, at least in local coordinates, which is evident from $$T(X,Y):= \nabla_XY -\nabla_YX -[X,Y] $$ and $$\nabla_{\partial_{i}}\partial_j = \sum_{k=1}^n \Gamma^k_{ij} \partial_k$$ where $\{\partial_1,...,\partial_n\}$ are the vector fields corresponding to some (local) coordinate system $\{x_1,...,x_n\}$. By definition, $T$ is the Torsion Tensor.

So... Is your Question:

  1. If this equation is ineed correct?
  2. If $T$ is indeed a Tensor?
  3. If this definition of $T$ is equivalent or related to some other definition?
  4. Something completly different?
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