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

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

2007-03-30

arxiv.org/abs/math/0703909v1

Mathematics

Differential Geometry

11 pages

Scientific paper

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

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