Short survey on the existence of slices for the space of Riemannian metrics

Abstract

We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more concise proof for the existence of slices.

Moduli space of non-negative sectional or positive Ricci curvature metrics on sphere bundles over spheres and their quotients

“…As a consequence of the Ebin slice theorem (see [17] and [11,Proposition 4.6]), this path lifts to a path in R C (M 8n−1 ) such that (0) = φ * 0 (g 0 ) and (1) = ψ * (φ * 1 (g 1 )) for some ψ ∈ Diff(M 8n−1 ). If ψ is orientation reversing, we can replace g 1 by its pullback under an orientation reversing diffeomorphism of M 8n−1 1 (the pullback of g 1 by this orientation reversing diffeomorphism still is a representative of [φ * 1 (g 1 )] in M C (M 8n−1 )), in order to Fix m 0 , m 1 ∈ Z such that |2k 0 + 1| = |2k 1 + 1| (where k i = k + 2 4n−2 m i • q n for i = 0, 1) and such that there exist orientation preserving diffeomorphisms i :…”

Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics

“…For a closed smooth manifold M , we denote by Met(M ) the space of smooth Riemannian metrics on M equipped with the C ∞ Whitney topology. For a survey on the properties of this space, the interested reader can consult, for example, [7], [14], [15], [18]. A Riemannian metric on M is pointwise strongly 1/4-pinched if for any point p ∈ M , the ratio of the maximal to the minimal sectional curvatures at p is strictly less than 4.…”

“…We end the proof by showing that φ g depends continuously on the metric g. Consider two metrics g 1 and g 2 on RP n close to each other with respect to C ∞ Whitney-topology. Since φ g 1 and φ g 2 are given by the exponential maps, as well as the Gram-Schmidt orthonormalization process with respect to g 1 and g 2 , then the isometry φ g 1 is close to the isometry φ g 2 in Diff(RP n ) with respect to the C ∞ Whitney topology (see [7,Section 2.6]).…”