Extension of Bochner-Lichnérowicz formula on spheres
type: Item
instance of: scholarly article
title: Extension of Bochner-Lichnérowicz formula on spheres
zbMATH Open document ID: 1088.58013
author: Dominique Bakry
author: Abdellatif Bentaleb
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2006-03-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73646
review text: Let \(\Delta\) be the usual scalar Laplacian on the standard sphere \(S^m\) in \(\mathbb R^{m+1}\). The authors construct a sequence of bilinear forms \(\text{R}_k\) which are positive and intrinsically associated with \(\Delta\) and which carry all the information on the spectrum. Let \(\lambda_i\) be the eigenvalues of \(\Delta\). If \(f\) is an eigenfunction associated to the eigenvalue \(\lambda\), and if \(g\) is an arbitrary smooth function, then \[ R_k(f,g)=\lambda(\lambda-\lambda_1)...(\lambda-\lambda_k)\int_{S^m}{fg\, dvol}\,. \] The authors extend to the full spectrum the usual Bochner-Lichnerowicz formula giving a lower bound for the non-zero eigenvalue; this bound is sharp on the sphere.
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 5012712
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: intrinsic bilinear operator
zbMATH Keywords: eigenvalue of the Laplacian
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
OpenAlex ID: W184404742
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DOI: 10.5802/AFST.1091
description: scientific article; zbMATH DE number 5012712