In this question here, I asked about computing the Levi-Civita connection matrix on the Hyperbolic Plane, defined as $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = \frac{1}{y^2}(dx^2+dy^2)$. The answer (which I believe is correct) was given as: $$\omega^i_j(X)=\begin{bmatrix} -\frac{X^2}{y} & -\frac{X^1}{y} \\ \frac{X^1}{y} & -\frac{X^2}{y} \end{bmatrix}\ .$$
But I have a proposition in my notes which reads:
Let $\xi$ be a smooth vector bundle equipped with an inner product $<\cdot ,\cdot>$ and a connection $\nabla$. Then $\nabla$ is compatible with the inner product if and only if in any local orthonormal frame $(s_i)$ the connection matrix $\omega$ is skew-symmetric.
The matrix $\omega$ above is clearly not skew-symmetric, and I'm not quite sure what to make of this apparent contradiction. It is certainly not true that there exists no Levi-Civita connection on the hyperbolic plane, so where am I going wrong?
As the proposition you quoted says, the connection forms have to be skew-symmetric with respect to any orthonormal frame. You computed the connection forms with respect to the coordinate frame $\{\partial/\partial x, \partial/\partial y\}$, which is not orthonormal for the hyperbolic metric.