Warped products with critical Riemannian metric
type: Item
instance of: scholarly article
title: Warped products with critical Riemannian metric
zbMATH Open document ID: 0854.53034
author: Byung Hak Kim
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-01-22T00:00:00Z
review text: Let \((B,g)\) and \((F,\overline{g})\) be two Riemannian manifolds and let \(f\) be a positive smooth function on \(B\). Then the warped product manifold \(M = B \times_f F\) is defined by the Riemannian metric \(\widetilde {g} = \pi^*(g)+ (f\circ \pi)^2 \sigma^* (\overline{g})\), where \(\pi\) and \(\sigma\) are the projections of \(B \times F\) onto \(B\) and \(F\), respectively. Let \(\widetilde{R}\), \(R\) and \(\overline{R}\) be the curvature tensors, and \(\widetilde{\gamma}\), \(\gamma\), and \(\overline{\gamma}\) the scalar curvatures of \(M\), \(B\), and \(F\), respectively. Consider the Riemannian functional \(H(\widetilde{\gamma}) =\int_M \widetilde{\gamma}^2 d\mu\) on \(M\), where \(d \mu\) is the volume element measured by \(\widetilde\gamma\). A critical point of \(H(\widetilde{\gamma})\) is called a critical Riemannian metric on \(M\). In this paper, the author proves that if \(\widetilde{\gamma}\) is a critical Riemannian metric on a warped product manifold \(M\), then \(M\) is the Riemannian product space or \(\widetilde{\gamma}\) is constant on \(M\). Also, it is shown that if \(\widetilde{\gamma} \neq 0\) is a critical Riemannian metric on \(M\), then \(\overline {\gamma}\) on \(F\) is a critical Riemannian metric if and only if \(F\) is Einstein.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58E11
zbMATH DE Number: 851724
zbMATH Keywords: warped product space
zbMATH Keywords: critical Riemannian metric
MaRDI profile type: Publication
full work available at URL: pjaa.71.117
OpenAlex ID: W2019720991
Wikidata QID: Q115219931
cites work: Quelques formules de variation pour une structure riemannienne
cites work: Q4720334
cites work: Manifolds of Negative Curvature
cites work: Q3312869
DOI: 10.3792/PJAA.71.117
description: scientific article; zbMATH DE number 851724