Poisson integrals for homogeneous, rank 1 Koszul manifolds
type: Item
instance of: scholarly article
title: Poisson integrals for homogeneous, rank 1 Koszul manifolds
zbMATH Open document ID: 0819.47028
published in: Journal of Functional Analysis
publication date: 1994-12-20T00:00:00Z
review text: Let \(X\) be a complex manifold which is homogeneous under the real analytic Lie group \(G\). From a \(G\)-invariant volume form \(Kdx\) on \(X\) is constructed a new form, the Koszul form, which is del-delbar of \(K\). If the Koszul form is non-degenerate then it gives \(X\) the structure of a pseudo-Kähler manifold. This then gives rise to a canonical second- order differential operator, the Laplace-Beltrami operator. An investigation of the properties of these forms leads to a complete solution of the eigenvalue problem for the Laplace-Beltrami operator for large classes of groups.
Mathematics Subject Classification ID: 47A75
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 32M10
zbMATH DE Number: 704490
zbMATH Keywords: complex manifold
zbMATH Keywords: homogeneous under the real analytic Lie group
zbMATH Keywords: \(G\)- invariant volume form
zbMATH Keywords: Koszul form
zbMATH Keywords: pseudo-Kähler manifold
zbMATH Keywords: second- order differential operator
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: eigenvalue problem for the Laplace-Beltrami operator
author: Richard Penney
reviewed by: David B. Gauld
MaRDI profile type: Publication
OpenAlex ID: W2087677656
DOI: 10.1006/JFAN.1994.1111
description: scientific article; zbMATH DE number 704490