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