Holonomy displacements in Hopf bundles over complex hyperbolic space and the complex Heisenberg groups

For the ``Hopf bundle'' $S^1\to S^{2n,1} \to {\mathbb H}^n$, horizontal lifts of simple closed curves are studied. Let $\gamma$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along $\gamma$ is given by $$V(\gamma)=e^{\lambda A(\gamma) i}$$ where $A(\gamma)$ is the area of the region on the surface $S$ surrounded by $\gamma$; $\lambda=1/2$ or 0 depending on whether $S$ is a complex submanifold or not. We also carry out a similar investigation for the complex Heisenberg group ${\mathbb R} \to {\mathcal H}^{2n+1} \to {\mathbb C}^n$.