Asymptotic curvature of \(\Theta\)-metrics
type: Item
instance of: scholarly article
title: Asymptotic curvature of \(\Theta\)-metrics
zbMATH Open document ID: 0891.53023
DOI: 10.1023/A:1006538612059
author: Matthew B. Stenzel
published in: Annals of Global Analysis and Geometry
publication date: 1998-07-06T00:00:00Z
review text: A \(\Theta\)-structure on a smooth compact manifold with boundary \(X\) is a projective class of one-forms \(\Theta\in C^{\infty}(\partial X, T^{\ast}X)\) which is non-vanishing when pulled back to \(\partial X\). Associated to the \(\Theta\)-structure there exists a Lie algebra of vector fields \({\mathcal V}_{\Theta}\), which lifts to the space of all smooth sections of a vector bundle \(^{\Theta}TX\). In the paper under review, given a \(\Theta\)-structure it is defined what is a \(\Theta\)-metric and it is studied the geometry of \(\Theta\)-metrics. The main result of the paper is the expression of the curvature tensor and the Ricci tensor of a \(\Theta\)-metric at boundary points, in terms of the metric tensor and invariants of the \(\Theta\)-structure. As a consequence it is obtained a relation of these invariants that ensures that the curvature tensor is Einstein at the boundary. Moreover, the author determines the structure of the curvature tensor of the induced metric on level sets of a defining function, in terms of the above mentioned invariants. Finally, some examples and applications are given: strictly pseudoconvex domains, Bergman metric, manifolds with contact boundary.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1038401
zbMATH Keywords: Bergman metric
zbMATH Keywords: manifold with boundary
zbMATH Keywords: \(\Theta\)-metric
zbMATH Keywords: \(\Theta\)-curvature
reviewed by: Raúl Ibañez
MaRDI profile type: Publication
full work available at URL: a:1006538612059
OpenAlex ID: W1668741347
description: scientific article; zbMATH DE number 1038401