Harmonic spaces and density functions
type: Item
instance of: scholarly article
title: Harmonic spaces and density functions
zbMATH Open document ID: 1462.53034
author: D. Kharzeev
published in: Results in Mathematics
publication date: 2020-07-14T00:00:00Z
review text: Let \(\mathcal{M=}\left( M,g\right) \) be an \(m\)-dimensional Riemannian manifold, \(P\in M\) and \(\exp_{P}:M_{p}\rightarrow M\) be the exponential map at \(P\). By fixing an orthonormal basis in \(M_{p}\), a local normal coordinates \(x^{1},x^{2},\dots,x^{m}\) centered at \(P\) via the exponential map is introduced. If \(\xi\) is a vector in \(M_{p}\) with coordinates \(\xi^{1},\xi^{2},...,\xi^{m} \), then the volume element in the local normal coordinates \(x^{1} ,x^{2},..,x^{m}\) is written as \(\widetilde{\Theta}_{\mathcal{M}}\left( P;\xi\right) d\xi\), where \(\widetilde{\Theta}_{\mathcal{M}}\left( P;\xi\right) \) is the density function. In polar coordinates, the density function is given by \(\Theta_{\mathcal{M}}\left( P;\xi\right) =r^{m-1}\) \(\widetilde{\Theta}_{\mathcal{M}}\left( P;\xi\right) \). A Riemannian manifold \(\mathcal{M}\) is called a harmonic space, if \(\Theta_{\mathcal{M} }\left( P;\xi\right) \) is a radial function \(\Theta_{\mathcal{M}}\left( P;r\right) \). The definition of harmonic spaces can be given in several equivalent ways (Theorem 1.3). Recall that the Jacobi operator \(J_{\mathcal{M} }\left( \xi\right) \eta\) is \(R_{\mathcal{M}}\left( \eta,\xi\right) \xi\). The authors consider harmonic spaces with the following density functions \[ \Theta_{\varepsilon}\left( m,k\right) \left( r\right) =\left\{ \begin{array} [c]{c} \sin\left( r\right) ^{m-1}\cos\left( r\right) ^{k}\text{ if } \varepsilon=+,\\ \sinh\left( r\right) ^{m-1}\cosh\left( r\right) ^{k}\text{ if } \varepsilon=-. \end{array} \right. \] In their main result (Theorem 1.8), the authors obtain estimates for averages of the eigenvalues of the (reduced) Jacobi operator and prove that when an equality holds in one of their estimates, then \(\mathcal{M}\) is locally isometric to a rank one symmetric space. The authors also prove estimates for individual eigenvalues (Theorem 1.9). Let \(\widetilde{\Theta}_{\mathcal{M}}\left( P;\xi\right) \sim1+\Sigma _{k=2}^{\infty}\mathcal{H}_{k}…
reviewed by: Igor G. Nikolaev
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 7220543
zbMATH Keywords: rank-one symmetric spaces
zbMATH Keywords: harmonic spaces
zbMATH Keywords: density function
MaRDI profile type: Publication
full work available at URL: s00025-020-01248-7
OpenAlex ID: W3040895369
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DOI: 10.1007/S00025-020-01248-7
description: scientific article; zbMATH DE number 7220543