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Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics. (English) Zbl 1292.58004
Let \(M\) be an \(n\)-dimensional compact closed manifold, and consider the infinite-dimensional space \(\mathcal{M}\) of all smooth Riemannian metrics on \(M\). The space \(\mathcal{M}\) is endowed with a natural \(L^2\)-type Riemannian structure, the Ebin metric \(g_E(h, k)|g := (h, k)_E := \int_M \text{tr}(g^{-1}hg^{-1}k)dV_g\), where \(g\in \mathcal{M}\), \(h, k\in T_g\mathcal{M}\), \(T_g\mathcal{M}\) may be identified with the space \(0''(S^2T^*M)\) of smooth symmetric (0,2)-tensor fields on \(M\).
Let \(f:\mathcal{M}\to \mathbb{R}\) be a twice continuously differentiable function, and consider the metric on \(\mathcal{M}\), \(g_f(h, k)|g := e^{2f(g)}g_E(h, k)|g\) conformal to the Ebin metric. There are distinguished metrics for which the conformal factors depend on the volume \(V_g := \mathrm{Vol}(M,g)\), i.e., metrics on \(\mathcal{M}\) of the form \(e^{2f(V_g)}g_E\), with \(f\) a smooth function on \(\mathbb{R}_+\), and mostly the metrics \(g_p := g_E/V^p\). An example of such a metric is the generalized Calabi metric (or sometimes the normalized Ebin metric), given by \(g_N := \frac{1}{V_g}g_E\), \(g\in \mathcal{M}\).
The purpose of this article is to study conformal deformations of the Ebin metric. The geometry of the generalization of Calabi’s metric is studied in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and the authors solve it explicitly by using a constant of the motion. The authors determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric.
This article includes the following sections: Introduction; Preliminaries; Conformal deformations of the Ebin metric; Geometry of the generalized Calabi metric; The distance functions and the metric completions; Remarks and open questions.

MSC:
58D17 Manifolds of metrics (especially Riemannian)
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58D25 Equations in function spaces; evolution equations
58E11 Critical metrics
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