The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group
type: Item
instance of: scholarly article
title: The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group
zbMATH Open document ID: 0694.58008
author: Daniel S. Freed
author: David Groisser
published in: Michigan Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let M be a compact, smooth, oriented n-manifold and consider the set \(M'\) of all Riemannian metrics on M. It is an open convex subset of the space \(\Gamma (S^ 2T^*M)\). It carries a canonical nondegenerate inner product which in this paper is denoted by \(>A,B>_ g=\int_{M}tr_ g(AB^ t)\mu (g)\) but which can more clearly be described by \(>A,B>_ g=\int_{M}tr(g^{-1}Ag^{-1}B)\mu (g)\) where each bilinear form on TM is viewed as a vector bundle homomorphism \(TM\to T^*M.\) In the first section the curvature of this Riemannian metric is computed by a rather complicated approach: The space M splits into \(M_{\mu}\times Vol(M)\), where \(M_{\mu}\) consists of all Riemannian metrics with fixed induced volume form \(\mu\) and where the second set corresponds roughly to a conformal class. For this splitting the equations of Weingarten, Gauss, Codazzy, and Ricci are used to derive the curvature. For Hilbert manifolds these equations make sense as proved in the book of \textit{W. Klingenberg}, ``Riemannian Geometry'' (1982; Zbl 0495.53036) which is not cited in the paper. But for this to work one should consider the Sobolev completion of the manifold M, which is not done in the paper. In the second section the geodesic equation is integrated explicitly, but without attention to the maximal domain of definition. The branch of arctan used should be described more carefully. The geodesic equation is an ordinary differential equation not involving any space derivatives. The third section is an attempt to compute the curvature of the induced metric on the quotient space \(M'/Diff^+(M)\) of the space of Riemannian metrics without isometries modulo the action of the group of orientation preserving diffeomorphisms. Since again the fundamental equations for submanifolds are used one should definitely take the Sobolev completion first if the computations should make sense. The stratification of the orbit space M/Diff(M) has been worked out by \textit{J.-P. Bourguignon} [Compos. Math. …
Mathematics Subject Classification ID: 58D17
Mathematics Subject Classification ID: 58B20
zbMATH DE Number: 4137038
zbMATH Keywords: manifold of Riemannian metrics
zbMATH Keywords: curvature
zbMATH Keywords: geodesic equation
zbMATH Keywords: quotient space
zbMATH Keywords: group of orientation preserving diffeomorphisms
Wikidata QID: Q115238991
reviewed by: Peter W. Michor
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004004
OpenAlex ID: W1974342677
DOI: 10.1307/MMJ/1029004004
description: scientific article; zbMATH DE number 4137038