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The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group. (English) Zbl 0694.58008
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 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 J.-P. Bourguignon [Compos. Math. 30, 1-41 (1975; Zbl 0301.58015)] which is not cited in the paper.
The reader should also consult the book of A. L. Besse [Einstein manifolds (Springer, Berlin) (1987; Zbl 0613.53001), Chapters 4 and 12] and the references therein. This is not cited in the paper.
Finally the reviewer wants to point out the paper of O. Gil-Medrano and the reviewer [The Riemannian manifold of all Riemannian metrics; Preprint (1989)] where for the same metric on the space of all Riemannian metrics on a not necessarily compact manifold the curvature, exponential mapping and all Jacobi fields are computed explicitly, in a simpler way, with more attention to the relevant questions of infinite dimensional topology and analysis.
Reviewer: P.Michor

58D17 Manifolds of metrics (especially Riemannian)
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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