How to know If the given space is torsionfree? I'm given a metric
$$ds^2=g_{\mu\nu}dx^\mu dx^\nu=\left(1-\frac{r_g}{r}\right)c^2dt^2-\left(1+\frac{r_g}{r}\right)dr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2$$
where $r_g$ is a constant.
The torsion tensor is
$$T^k_{ij}\equiv \Gamma^k_{ij}-\Gamma^k_{ji}.$$
But the formula that I'm using that is
$$\Gamma^m_{ik}=\frac{1}{2}g^{ml}\left(\frac{\partial g_{li}}{\partial x^k}+\frac{\partial g_{lk}}{\partial x^i}-\frac{\partial g_{ik}}{\partial x^l}\right)$$
already assumes that the torsion tensor is zero. Can anyone help me out, How to see if $T^k_{ij}$ is zero or not?
 A: Both torsion and curvature are properties of a connection, not an intrinsic property of a metric manifold itself.  For a given manifold $\mathcal M$ with pseudo-Riemannian metric $g$, you could equally well equip it with a connection $\nabla$ which has torsion or a connection $\nabla'$ which does not, so the metric components are not going to help you here.
Of the infinity of possible choices of connection for a given $(\mathcal M,g)$, there is one unique connection $\nabla^{LC}$ which is both metric-compatible $(\nabla^{LC} g = 0)$ and torsion-free; this is the Levi-Civita connection, and it is the choice that is made in standard GR.  Imposing both of those constraints yields the expression for the connection coefficients written in the OP,
$$\left(\Gamma^{LC}\right)^m_{ik}=\frac{1}{2}g^{ml}\left(\frac{\partial g_{li}}{\partial x^k}+\frac{\partial g_{lk}}{\partial x^i}-\frac{\partial g_{ik}}{\partial x^l}\right)$$
When we talk about torsion and curvature in the context of standard GR, we're talking about the torsion and curvature of the Levi-Civita connection.  Different choices for connection would yield different torsion and curvature. For example, the coefficients of any metric-compatible connection can be expressed in terms of those of the Levi-Civita connection via
$$\Gamma^m_{ik} = \left(\Gamma^{LC}\right)^m_{ik} + K^m_{ik}$$
where $K$ is the contorsion tensor.  As a more interesting example, one might consider teleparallel gravity which equips $(\mathcal M,g)$ with the Weitzenbock connection rather than the Levi-Civita connection; this connection is curvature-free but does possess generally non-vanishing torsion, and it is this torsion which is manifested in the effects of gravity.
