I am trying to understand connections in fibre bundles. I thought of the following problem:
Let $\Gamma$ be the discrete group generated by
\begin{pmatrix} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 6 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
Then $\Gamma$ acts properly discontinuously on the Heisenberg group $H$ with compact quotient. The extension $0\to C(\Gamma) \to \Gamma \to \Gamma/C(\Gamma) \to 0$ shows that $H/\Gamma$ is a circle bundle over the torus.
The numbers $3,4$ seem to determine a Riemannian structure on the torus, as when one quotients $\mathbb R^2$ by $(3\mathbb Z,4\mathbb Z)$ & similarly $6$ determines a Riemannian structure on the circle.
The standard flat connection on $H \cong \mathbb R^2 \times \mathbb R$ should induce a connection on the quotient $H / \Gamma$. Is there a way to see the numbers $3,4,6$ explicitly in the connection form of the quotient?