# Torsion and Non-metricity Tensor on a Surface

In differential geometry of surfaces, how can one define a non-zero Torsion tensor? It seems that the connection you provide has always to be symmetric since, by definition, $$\Gamma^{\gamma}_{\alpha\beta}\equiv \mathbf{a}^{\gamma}\cdot\mathbf{a}_{\alpha,\beta}=\mathbf{a}^{\gamma}\cdot\mathbf{r}_{,\alpha\beta}=\mathbf{a}^{\gamma}\cdot\mathbf{r}_{,\beta\alpha}=\Gamma^{\gamma}_{\beta\alpha},$$ where $\mathbf{r}:U\to\mathbb{R}^3$, $U\subset\mathbb{R}^2$, is an embedded $C^3$ surface with parametrization $(\theta^1,\theta^2)\in U$, $\mathbf{a}_\alpha\equiv\mathbf{r}_{,\alpha}$ are the tangent vectors to the coordinate curves $\theta^\alpha$, $\alpha=\{1,2\}$, and $\mathbf{a}^\gamma$ is the covector of $\mathbf{a}_\alpha$.

This definition also implies that the connection is Levi-Civita, i.e. metric compatible: $$\Gamma^{\gamma}_{\alpha\beta}=\frac{1}{2}a^{\gamma\lambda}(a_{\beta\lambda,\alpha}+a_{\gamma\alpha,\beta}-a_{\alpha\beta,\lambda}),$$ which means that the covariant derivative of the metric tensor will be automatically zero. So there is no non-zero Non-metricity Tensor either.

Existence of non-zero Torsion tensor and Non-metricity tensor is important in studies of defects in two-dimensional crystals because in continuum model, they represent certain defect densities.

For a linear connection $\nabla : \mathfrak{X}(M)\times \mathfrak{X}(M) \rightarrow \mathfrak{X}(M)$, given vector fields $X,Y \in \mathfrak{X}(M)$ torsion tensor is $(2,0)$ tensor field given by $$\tau(X,Y) = \nabla_X(Y)-\nabla_Y(X) -[X,Y]$$ The symmetry of the connection is equivalent to $\tau (X,Y) = 0$ for all $X,Y \in \mathfrak{X}(M)$. I don't exactly understand your notation, but what you referred as " by definition " is probably true only for Christoffel symbols corresponding to Levi-civita connection which is indeed symmetric.