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 ${ \mathbb R} \to { \mathcal H}^{ 2n+1} \to { \mathbb C}^n$.
Categories: math.DG
