This a problem from Loring Tu, 'Differential Geometry':
Problem: The Poincare disk is the open unit disk $$ \mathbf{D} = \left\{ z = x + iy \in \mathbb{C} \mid | z| < 1 \right\} $$ in the complex plane with metric $$ \langle , \rangle_z = \frac{ 4 (dx \otimes dx + dy \otimes dy)}{ (1- |z|^2)^2}. $$ An orthonormal frame for $\mathbf{D}$ is $$e_1 = \frac{1}{2} (1 – |z|^2) \partial_x, \qquad e_2 = \frac{1}{2} (1 – |z|^2) \partial_y. $$ Find the connection matrix $ \omega = [\omega_j^{i}]$ relative to the orthonormal frame $e_1, e_2$ of the Riemannian connection $\nabla$ on the Poincare disk (Hint: first find the dual frame $\theta^{1}, \theta^{2}$ and then solve the first structural equation).
Attempt: The dual frame satisfies $\theta^{i} (e_j ) = \delta_{j}^{i}$. So that means $$ \theta^{1} = \frac{2}{ 1 – |z|^2} dx, \qquad \theta^{2} = \frac{2}{ 1 – |z|^2} dy. $$
We have zero torsion so the first structural equation reads $$ d \theta^{i} + \omega_j^{i} \wedge \theta^j = 0 $$ for $ i = 1, 2$ and summation is implied. I calculated $$ d \theta^1 = \partial_y \left( \frac{2}{ 1 – x^2 – y^2} \right) dy \wedge dx = \frac{ – 4y}{ (1 – x^2 – y^2)^2} dx \wedge dy $$ and $$ d \theta^2 = \frac{ 4 x}{ (1 – x^2 – y^2)^2} dx \wedge dy. $$
Since we have an orthonormal basis, I know that the connection matrix should be skew-symmetric. So this means $\omega^{1}_{1} = \omega_2^{2} = 0$. Then I need to solve e.g. $$ d \theta^1 + \omega_2^{1} \wedge \theta^2 =0$$ for $\omega_2^{1}$.
Since I have a dual frame, I can always expand (right?) $$ \omega_2^{1} = a_1 \theta^1 + a_2 \theta^2 $$ for some coefficients $a_1$ and $a_2$. So the above equation becomes $$ d\theta^1 + a_1 \theta^1 \wedge \theta^2 = 0. $$ In other words : $$ – \frac{4y}{ (1 – |z|^2)^2} dx \wedge dy + \frac{ 4 a_1}{ (1 – |z|^2)^2} dx \wedge dy = 0 $$ so that $a_1 = y$. So $\omega_2^{1} = y \theta^1$.
Similarly, I found $$ \omega_1^{2} = x \theta^2. $$ But this doesn't seem correct to me since I should have $\omega_1^{2} = – \omega_2^{1}$?
Where am I going wrong? Thanks for any help.
Best Answer
Your mistake is the line "so $\omega^1_2 = y\theta^1$." The structure equation for $d\theta^1$ allows the $\theta^2$ component of $\omega^1_2$ to be arbitrary (since $\theta^2\wedge\theta^2=0$), and similarly for the $\theta^1$ component of $\omega^2_1$. So the two structure equations and skew-symmetry of the connection form will in fact determine $\omega^1_2 = -\omega^2_1$ uniquely.