Consider $S^{2}$, the 2-sphere in $\mathbb{R}^{3}$. Let $(\theta, \phi)$ be the coordinates of parametrization of $S^{2}$ given by $$ X(\theta, \phi)=(\sin{\theta}\cos{\phi}, \sin{\theta}\sin{\phi}, \cos{\theta}) \ \ 0 \leq \theta \leq \pi, 0 \leq \phi \leq 2\pi $$.
We have that
$$ X_{\theta}=(\cos{\theta}\cos{\phi}, \cos{\theta}\sin{\phi}, -\sin{\theta})
\ \ e \ \ X_{\phi}=(-\sin{\theta}\sin{\phi}, \sin{\theta}\cos{\phi}, 0)$$
and
$$
g_{S^{2}}= \begin{bmatrix}
1 & 0 \\
0 & \sin^{2}{\theta}
\end{bmatrix}
$$ Then
$$
\text{volume form} = \sqrt{|g_{S^{2}}|}=\sin{\theta} d\theta\wedge d\phi
$$
I would like calculate the connection matrix for $S^{2}$. I found
$$
\omega_{\theta} = \begin{bmatrix}
cos\theta dx^{\theta} & 0 \\
0 & 0
\end{bmatrix}
$$
e
$$
\omega_{\phi} = \begin{bmatrix}
0 & \cot{\theta}dx^{\phi} \\
-sin{\theta}\cos{\theta}dx^{\phi} & 0
\end{bmatrix}
$$
It is correct? I used Christoffel Symbols. Can I used volume form for calculate the matrix of 1-forms? Hint?