Math 2.0

Global analysis, analysis on manifolds Manifolds of mappings and moduli spaces Theorem

Ebin's slice theorem

Builds on Smooth manifoldRiemannian metricDiffeomorphism groupStabiliserIsometryManifold of smooth mappings

Statement

Let MM be a compact smooth manifold without boundary. Let ℳ\mathcal{M} be the space of smooth Riemannian metrics on MM, an open subset of the Fréchet space of smooth symmetric 22-tensors. Let 𝒟\mathcal{D} be the diffeomorphism group of MM, acting on ℳ\mathcal{M} on the right by pullback, (γ,η)↦η∗γ(\gamma, \eta) \mapsto \eta^* \gamma. For γ∈ℳ\gamma \in \mathcal{M}, let Iγ⊆𝒟I_\gamma \subseteq \mathcal{D} be its isometry group, which is the stabiliser of γ\gamma. Then there is a submanifold SγS_\gamma of ℳ\mathcal{M} containing γ\gamma, the slice at γ\gamma, such that:

  1. If η∈Iγ\eta \in I_\gamma, then η∗Sγ=Sγ\eta^* S_\gamma = S_\gamma.
  2. If η∈𝒟\eta \in \mathcal{D} and η∗Sγ∩Sγ≠∅\eta^* S_\gamma \cap S_\gamma \neq \emptyset, then η∈Iγ\eta \in I_\gamma.
  3. There are a neighbourhood UU of the identity coset in 𝒟/Iγ\mathcal{D} / I_\gamma and a local section χ ⁣:U→𝒟\chi \colon U \to \mathcal{D} of the projection 𝒟→𝒟/Iγ\mathcal{D} \to \mathcal{D} / I_\gamma. For these, the map U×Sγ→ℳU \times S_\gamma \to \mathcal{M}, (u,s)↦χ(u)∗s(u, s) \mapsto \chi(u)^* s, is a homeomorphism onto a neighbourhood of γ\gamma.

Remarks

The theorem is due to Ebin. The tangent space to the orbit of γ\gamma consists of the Lie derivatives LXγL_X \gamma of γ\gamma along vector fields XX. Its L2L^2-orthogonal complement is the space of symmetric 22-tensors hh with δγh=0\delta_\gamma h = 0, where δγ\delta_\gamma is the divergence. The slice is the image of a small ball in this complement under the exponential map of an IγI_\gamma-invariant metric on ℳ\mathcal{M}. The analysis is carried out on Sobolev completions of ℳ\mathcal{M} and 𝒟\mathcal{D}, which are Hilbert manifolds, and then passed to the smooth case. Compactness of IγI_\gamma, which is a compact Lie group because MM is compact, is used to average and to make the slice invariant. It is the infinite-dimensional analogue of the slice theorem for actions of compact Lie groups on finite-dimensional manifolds.

Example: suppose that γ\gamma has trivial isometry group. Then near γ\gamma the action is free, and a neighbourhood of the class of γ\gamma in ℳ/𝒟\mathcal{M} / \mathcal{D} is homeomorphic to the slice SγS_\gamma.

Going deeper

For a student

The slice theorem is the basic tool for describing the moduli space of Riemannian metrics, sometimes called superspace in general relativity. It shows that this space splits into pieces according to the isometry group of the metric. Near a metric whose only isometry is the identity, the space is locally modelled on the slice. The same pattern reappears for connections modulo gauge: an infinite-dimensional group acts with a slice built from an elliptic gauge condition. There the Coulomb gauge plays the role of the divergence-free condition.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectExponential map of a Lie group, concept, no evidence levelExponential map of a …conceptFréchet manifold, concept, no evidence levelFréchet manifoldconceptFunction, concept, no evidence levelFunctionconceptGroup, concept, no evidence levelGroupconceptIsometry, concept, no evidence levelIsometryconceptLie group, concept, no evidence levelLie groupconceptManifold of smooth mappings, concept, no evidence levelManifold of smooth ma…conceptRiemannian metric, concept, no evidence levelRiemannian metricconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptVector field, concept, no evidence levelVector fieldconceptGroup action, definition, no evidence levelGroup actiondefinitionDiffeomorphism, concept, no evidence levelDiffeomorphismconceptStabiliser, definition, no evidence levelStabiliserdefinitionDiffeomorphism group, example, no evidence levelDiffeomorphism groupexampleEbin's slice theorem, statement, no evidence levelEbin's slice theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Exponential map of a Lie group concept
  3. Fréchet manifold concept
  4. Function concept
  5. Group concept
  6. Isometry concept
  7. Lie group concept
  8. Manifold of smooth mappings concept
  9. Riemannian metric concept
  10. Smooth manifold concept
  11. Total derivative concept
  12. Function of several variables concept
  13. Vector-valued function concept
  14. Group homomorphism definition
  15. Subgroup definition
  16. Symmetric group definition
  17. Differentiable function of several variables concept
  18. Vector field concept
  19. Group action definition
  20. Diffeomorphism concept
  21. Stabiliser definition
  22. Diffeomorphism group example
  23. Ebin's slice theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Slice_theorem_(differential_geometry)
About this page
TagAZF9TXV5C0
URIm2://AZF9TXV5C0
Kindstatement, theorem
WritingW0 stub
MSC58D17, 58D19, 58D05
StatusThis copy of the site has no build data. Run m2 status AZF9TXV5C0.
AboutRiemannian metric, Diffeomorphism group
UsesSmooth manifold, Riemannian metric, Diffeomorphism group, Stabiliser, Isometry, Manifold of smooth mappings

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