For the upper half-plane $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = \frac{1}{y^2}(dx^2+dy^2)$, I computed the Christoffel symbols as follows: $$\begin{align}\Gamma^1_{12}&=\Gamma^1_{12}=-\frac{1}{y}\\\Gamma^2_{11}&=\frac{1}{y}\\\Gamma^2_{22}&=-\frac{1}{y}\end{align}$$
Then using the relation between the connection matrix and the Christoffel symbols $\omega^k_j=\Gamma^k_{ij}dx^i$, I computed $$\begin{align} \omega^1_1 &=-\frac{1}{y}dy\\ \omega^1_2 &=-\frac{1}{y}dx\\ \omega^2_1 &=\frac{1}{y}dx\\ \omega^2_2 &=-\frac{1}{y}dy \end{align}$$ I now have two questions:
- To get the matrix $\omega$ proper, do I now need to contract an index with the metric?
- How do I actually carry out a computation of $\nabla_X(S)$ for $X\in\Gamma(T\mathbb{H}^2),\ S\in\Gamma(\xi)$? It's clear to me that I should just stick the column vector $S$ to the right of the matrix $\omega$ and multiply in the obvious way, but how do I "plug in" the vector field $X$? An example would be really helpful.
Thanks!
I have some perplexities concerning the equation linking $\omega$ to $\Gamma$. I get \begin{equation} \omega^k_j(X)\partial_k = \nabla_X\partial_j = \nabla_{X^i\partial_i}\partial_j=X^i\nabla_{\partial_i}\partial_j=X^i\Gamma^k_{ij}\partial_k \end{equation} hence $\omega^k_j(X)=X^i\Gamma^k_{ij}$, which gives the dependence of $\omega$ from the vector field. Then the components I get are \begin{bmatrix} -\frac{X^2}{y} & -\frac{X^1}{y} \\ \frac{X^1}{y} & -\frac{X^2}{y} \end{bmatrix} and if you plug in some vector field $Y=(Y^1,Y^2)$ you need to exhaust the $j$ index: \begin{equation} \omega(Y)=(-\frac{X^2}{y}Y^1-\frac{X^1}{y}Y^2,\frac{X^1}{y}Y^1 -\frac{X^2}{y}Y^2) \end{equation} I hope this is correct & helps!