Holonomy displacements in the Hopf bundles over \(\mathbb C H^n\) and the complex Heisenberg groups (Q2898895)

The authors study horizontal lifts of simple closed curves for the bundle NEWLINE\[NEWLINEU(1)=S^1\rightarrow S^{2n,1} \rightarrow U(1,n)/(U(1)\times U(n))=\mathbb{C}H^n,NEWLINE\]NEWLINE where NEWLINE\[NEWLINES^{2n,1}=\{(z_0,\dotsc,z_n)\in \mathbb{C}^{n+1}\,:\, -|z_0|^2+\sum_{i=1}^n|z_i|^2=-1\}\cong U(1,n)/U(n).NEWLINE\]NEWLINE For a simple closed curve \(\gamma(t)\), \(0 \leq t \leq 1\), on \(\mathbb{C}H^n\), the holonomy displacement \(V(\gamma)\in S^1\) along \(\gamma\) is defined as follows. Let \(\widetilde{\gamma}(t)\) be the horizontal lift of \(\gamma\), then \(\widetilde{\gamma}(1)=V(\gamma) \widetilde{\gamma}(0)\). The main result (Theorem 3.1) states that, if \(\gamma\) is a piecewise smooth, simply 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 \(\gamma\) and \(\lambda=1/2\) or \(0\) depending on whether \(S\) is a complex submanifold or not. A similar result is obtained for the complex Heisenberg group \(\mathbb{R}\rightarrow \mathcal{H}^{2n+1}\rightarrow\mathbb{C}^n\) (Theorem 4.1).