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For the \Hopf bundle" S 1 ! S 2n;1 ! CH n , horizontal lifts of simple closed curves are studied. Let be a piecewise smooth, simple closed curve on a complete totally geodesic surface S in the base space. Then the holonomy displacement along is given by V ( ) = e A ( )i ; where A( ) is the area of the region on the surface S surrounded by ; = 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 R ! H 2n+1 ! C n .

HOLONOMY DISPLACEMENTS IN THE HOPF BUNDLES OVER $\mathcal{C}$HnAND THE COMPLEX HEISENBERG GROUPS