The Laplacian and the heat kernel acting on differential forms on spheres
type: Item
instance of: scholarly article
title: The Laplacian and the heat kernel acting on differential forms on spheres
zbMATH Open document ID: 1219.58010
author: Masayoshi Nagase
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2010-03-30T00:00:00Z
review text: Consider the identity \(S^n\equiv SO(n+1)/SO(n)\) defined by the canonical action of \(SO(n+1)\) on the \(n\)-sphere. Let \(T_eSO(n)\equiv\mathfrak{g}=\mathfrak{so}(n+1)\) be endowed with the inner product \((\cdot,\cdot)=-B(\cdot,\cdot)\), where \(B\) is the Killing form. Let \(\mathfrak{k}=\mathfrak{so}(n)\subset\mathfrak{g}\), and let \(\mathfrak{m}\) denote its orthogonal complement. Thus \(T_{[e]}S^n\equiv\mathfrak{m}\). Let \(E_{i,j}\) be the \((n+1)\times(n+1)\)-matrix of which the \((i,j)\)-entry is \(1\) and the other entries are \(0\). Then \(e_{i,j}=(2(n-1))^{-1/2}(E_{i,j}-E_{j,i})\) (\(1\leq i<j\leq n\)) is a canonical orthonormal frame of \(\mathfrak{g}\). This frame is also denoted by \((e_1,\dots,e_{n(n+1)/2})\), or by \((e_{\mathfrak{m}1},\dots,e_{\mathfrak{m}n},e_{\mathfrak{k}1}\dots,e_{\mathfrak{k}n(n-1)/2})\) according to the decomposition \(\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{k}\). The corresponding dual frame of \(\mathfrak{g}^*\) is denoted by \((e^1,\dots,e^{n(n+1)/2})\), or \((e^{\mathfrak{m}1},\dots,e^{\mathfrak{m}n},e^{\mathfrak{k}1}\dots,e^{\mathfrak{k}n(n-1)/2})\). The superindex or subindex \(L\) is added to denote the corresponding left invariant vector fields or \(1\)-forms on \(SO(n+1)\). Then the Laplacian of \(SO(n+1)\) acting on functions is given by \(\Delta_0^{SO(n+1)}=-\sum_ie_i^Le_i^L\). The paper studies the lift of the Laplacian \(\Delta^{S^n}\), acting on differential forms, by the Riemannian submersion \(\pi:SO(n+1)\to S^n\). Let \(\tilde\omega=\sum_\alpha e^{\mathfrak{m}\alpha}_L\cdot\Omega_\alpha\) be the lift of a differential form \(\omega\), where the usual multi-index notation is used with \(\alpha=(\alpha_1,\dots,\alpha_{|\alpha|})\), and the coefficients \(\Omega_\alpha\) are functions on \(SO(n+1)\). The first main result of the paper states that \(\widetilde{\Delta^{S^n}\omega}=\sum_\alpha e^{\mathfrak{m}\alpha}_L\cdot\Delta_0^{SO(n+1)}\Omega_\alpha\). This is satisfactory because the study of the heat operat…
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58J37
Mathematics Subject Classification ID: 58A10
zbMATH DE Number: 5687944
zbMATH Keywords: sphere
zbMATH Keywords: Laplacian
zbMATH Keywords: heat kernel
reviewed by: Jesús A. Álvarez López
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1264084500
OpenAlex ID: W2049798232
cites work: Volumes of compact symmetric spaces
cites work: Noyau de diffusion sur les espaces homogènes compacts
cites work: Q3344779
cites work: Q3897665
cites work: Q3331943
cites work: Spectra and eigenforms of the Laplacian on \(S^n\) and \(P^n(C)\)
cites work: The spectra of 1-forms on simply connected compact irreducible Riemannian symmetric spaces
cites work: The spectra of 1-forms on simply connected compact irreducible Riemannian symmetric spaces. II
cites work: Lie groups beyond an introduction
cites work: Twistor spaces and the general adiabatic expansions
cites work: Q3100282
cites work: The heat equation on compact Lie group
cites work: Analytic torsion of space forms of certain compact symmetric spaces
DOI: 10.2748/TMJ/1264084500
description: scientific article; zbMATH DE number 5687944