The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group.

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

“…These formulas have been independently derived by Freed and Groisser [49] and Michor and Gil-Medrano [52]. A similar result is also available for the metric G Φ with Φ(Vol) = 1 Vol ; see [33].…”

“…This metric has been introduced in [43] and is also known as the Ebinmetric. Its geodesic equation and curvature have been calculated in [49,52], and the induced distance function and metric completion have been studied by Clarke [28][29][30][31]. Similar to Riemannian metrics on immersions, Sobolev metrics of higher order and almost local metrics can be defined using a (pseudo differential) operator field L acting on the tangent space of Met(M ).…”

“…4]. Subsequently Freed and Groisser [49] and Michor and Gil-Medrano [52] computed the geodesic equation and found explicit solution formulas. The geodesic equation for higher order Sobolev type metrics and Scalar curvature metrics can be found in [15] and for volume weighted metrics in [15,33].…”

The Superspace of geometrodynamics

“…For λ = 0 the infinite-dimensional geometry of Riem(Σ) has been studied in [27]. They showed that all curvature components involving one or more pure-trace directions vanish and that the curvature tensor for the trace-free directions is given by (now making use of the natural Lie-algebra structure of T Σ ⊗ T * Σ)…”

Perturbations of the metric in Seiberg-Witten equations

“…General facts on vector fields of spaces of sections of fiber bundles (see [6,Appendix]) imply that the curvature Ω of the bundle Ξ -Met(M ) is pointwise the curvature of the bundle…”