Spectrum and holonomy of the line bundle over the sphere
type: Item
instance of: scholarly article
title: Spectrum and holonomy of the line bundle over the sphere
zbMATH Open document ID: 0538.53039
DOI: 10.1007/BF01174183
author: Ruishi Kuwabara
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173505
review text: 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 \(S*S^ n\).
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 3855921
zbMATH Keywords: Hermitian line bundle
zbMATH Keywords: linear connection
zbMATH Keywords: Bochner-Laplacian
zbMATH Keywords: spectrum
zbMATH Keywords: holonomy
zbMATH Keywords: closed geodesic
MaRDI profile type: Publication
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OpenAlex ID: W2044850211
description: scientific article; zbMATH DE number 3855921