HOLONOMY DISPLACEMENTS IN THE HOPF BUNDLES OVER CHn AND THE COMPLEX HEISENBERG GROUPS

For the "Hopf bundle" S-1 -> S-2n,S-1 -> CHn, 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 R -> H2n+1 -> C-n.