# Christoffel symbols and torsion of the connection

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

• $T^k_{ij}$ measures how much the metric $g_{lk}$ is an-holonomic, i.e. the tangent basis is far from a coordinated one – janmarqz Nov 1 '18 at 18:05

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.
2. If $$T$$ is indeed a Tensor?
3. If this definition of $$T$$ is equivalent or related to some other definition?