The metric geometry of the manifold of Riemannian metrics over a closed manifold.

*(English)*Zbl 1213.58007Let \(\mathcal{M}\) be the infinite dimensional Fréchet manifold of all \(C^\infty\) Riemannian metrics on a fixed closed, finite-dimensional, orientable manifold \(M\). This manifold has a natural weak Riemannian metric called the \(L^2\) metric because it induces an \(L^2\)-type scalar product on the tangent spaces of \(\mathcal{M}\). The main result of the paper is that this weak metric induces a metric space structure on \(\mathcal{M}\).

This result is non-trivial because, a priori, the distance function induced by a weak metric on a Fréchet manifold is only a pseudometric, meaning that the distance between some points can be zero. Thus, in their work motivated by research in computer vision, P. W. Michor and D. Mumford found examples of weak Riemannian metrics on Fréchet manifolds for which the distance between any two points is zero [Doc. Math., J. DMV 10, 217–245 (2005; Zbl 1083.58010)].

A motivation for studying the metrics on the manifold \(\mathcal{M}\) comes from Teichmüller theory. Indeed, if \(M\) is a Riemann surface of genus greater than one, then the Techmüller space \(\mathcal{T}\) of \(M\) is a quotient of the submanifold of \(\mathcal{M}\) which consists of hyperbolic metrics. It can be shown that the \(L^2\) metric on \(\mathcal{M}\) descends to \(\mathcal{T}\) and is isometric (up to a constant scalar factor) to the Weil-Petersson metric on \(\mathcal{T}\).

An important ingredient of the proof of the main result is an explicit relation between the induced distance between two metrics in \(\mathcal{M}\) and the corresponding volume forms on the initial manifold \(M\).

This result is non-trivial because, a priori, the distance function induced by a weak metric on a Fréchet manifold is only a pseudometric, meaning that the distance between some points can be zero. Thus, in their work motivated by research in computer vision, P. W. Michor and D. Mumford found examples of weak Riemannian metrics on Fréchet manifolds for which the distance between any two points is zero [Doc. Math., J. DMV 10, 217–245 (2005; Zbl 1083.58010)].

A motivation for studying the metrics on the manifold \(\mathcal{M}\) comes from Teichmüller theory. Indeed, if \(M\) is a Riemann surface of genus greater than one, then the Techmüller space \(\mathcal{T}\) of \(M\) is a quotient of the submanifold of \(\mathcal{M}\) which consists of hyperbolic metrics. It can be shown that the \(L^2\) metric on \(\mathcal{M}\) descends to \(\mathcal{T}\) and is isometric (up to a constant scalar factor) to the Weil-Petersson metric on \(\mathcal{T}\).

An important ingredient of the proof of the main result is an explicit relation between the induced distance between two metrics in \(\mathcal{M}\) and the corresponding volume forms on the initial manifold \(M\).

Reviewer: Mikhail Belolipetsky (Durham)

##### MSC:

58D17 | Manifolds of metrics (especially Riemannian) |

58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

##### Keywords:

manifold of Riemannian metrics; superspace; manifold of Riemannian structures; \(L^{2}\) metric##### References:

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