Spectrum and holonomy of the line bundle over the sphere
Let E be a Hermitian line bundle over the n-sphere equipped with the standard metric. Consider a linear connection \(\tilde d\) on E compatible with the Hermitian structure. Then, in a natural way there can be defined a second order, non-negative, elliptic differential operator L (called the Bochner-Laplacian) which operates on cross-sections of E. If \(\tilde d\) is flat, L is reduced to the Laplace-Beltrami operator \(\Delta\). It is well-known that the spectrum of \(\Delta\) consists of eigenvalues \(\lambda_ k=k(k+n-1)\), \(k=0,1,2,..\). This paper clarifies how eigenvalues \(\{\mu_ j\}^{\infty}_{j=0}\) of L distribute in \([\lambda_ k,\lambda_{k+1}]\) under the influence of the linear connection \(\tilde d\). Let \(J=[a,b]\subset [0,1]\) and let \(J_ k=[\lambda_ k+aC_ k\), \(\lambda_ k+bC_ k] (C_ k=\lambda_{k+1}- \lambda_ k)\). Then, roughly speaking, the result is that \[ \#\{j;\quad \mu_ j\in J_ k\}/\#\{j;\quad \mu_ j\in [\lambda_ k,\lambda_{k+1}]\}\sim vol\{\bar Q^{-1}_{\tilde d}\quad(\exp(2\pi iJ))\}/vol\;S*S^ n\quad(k\to \infty), \] where \({\bar Q}_{\tilde d}\) is the map of \(S*S^ n\) into \(S^ 1=\{e^{2\pi i\theta}\); 0\(\leq \theta \leq 1\}\) defined by the holonomy of \(\tilde d\) along the closed geodesic of \(S^ n\) whose initial data is a given point of \(S*S^ n\).
scientific article; zbMATH DE number 3855921