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- Title Pages
- Epigraph
- Frontispiece
- Preface
- 1 Introduction
- 2 The Cauchy problem in general relativity
- 3 The topology of the universe
- 4 Notions of proximity to spatial homogeneity and isotropy
- 5 Observational support for the standard model
- 6 Concluding remarks
- 7 Main results
- 8 Outline, general theory of the Einstein–Vlasov system
- 9 Outline, main results
- 10 References to the literature and outlook
- 11 Basic analysis estimates
- 12 Linear algebra
- 13 Coordinates
- 14 Function spaces for distribution functions I: local theory
- 15 Function spaces for distribution functions II: the manifold setting
- 16 Main weighted estimate
- 17 Concepts of convergence
- 18 Uniqueness
- 19 Local existence
- 20 Stability
- 21 The Vlasov equation
- 22 The initial value problem
- 23 Existence of a maximal globally hyperbolic development
- 24 Cauchy stability
- 25 Spatially homogeneous metrics, symmetry reductions
- 26 Criteria ensuring global existence
- 27 A potential with a positive non-degenerate local minimum
- 28 Approximating perfect fluids with matter of Vlasov type
- 29 Background material
- 30 Estimating the Vlasov contribution to the stress energy tensor
- 31 Global existence
- 32 Asymptotics
- 33 Proof of the stability results
- 34 Models, fitting the observations, with arbitrary closed spatial topology
- A Examples of pathological behaviour of solutions to nonlinear wave equations
- B Quotients and universal covering spaces
- C Spatially homogeneous and isotropic metrics
- D Auxiliary computations in low regularity
- E The curvature of left invariant metrics
- F Comments concerning the Einstein–Boltzmann system
- References
- Index
- [UNTITLED]

# (p.695) E The curvature of left invariant metrics

# (p.695) E The curvature of left invariant metrics

- Source:
- On the Topology and Future Stability of the Universe
- Publisher:
- Oxford University Press

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- Title Pages
- Epigraph
- Frontispiece
- Preface
- 1 Introduction
- 2 The Cauchy problem in general relativity
- 3 The topology of the universe
- 4 Notions of proximity to spatial homogeneity and isotropy
- 5 Observational support for the standard model
- 6 Concluding remarks
- 7 Main results
- 8 Outline, general theory of the Einstein–Vlasov system
- 9 Outline, main results
- 10 References to the literature and outlook
- 11 Basic analysis estimates
- 12 Linear algebra
- 13 Coordinates
- 14 Function spaces for distribution functions I: local theory
- 15 Function spaces for distribution functions II: the manifold setting
- 16 Main weighted estimate
- 17 Concepts of convergence
- 18 Uniqueness
- 19 Local existence
- 20 Stability
- 21 The Vlasov equation
- 22 The initial value problem
- 23 Existence of a maximal globally hyperbolic development
- 24 Cauchy stability
- 25 Spatially homogeneous metrics, symmetry reductions
- 26 Criteria ensuring global existence
- 27 A potential with a positive non-degenerate local minimum
- 28 Approximating perfect fluids with matter of Vlasov type
- 29 Background material
- 30 Estimating the Vlasov contribution to the stress energy tensor
- 31 Global existence
- 32 Asymptotics
- 33 Proof of the stability results
- 34 Models, fitting the observations, with arbitrary closed spatial topology
- A Examples of pathological behaviour of solutions to nonlinear wave equations
- B Quotients and universal covering spaces
- C Spatially homogeneous and isotropic metrics
- D Auxiliary computations in low regularity
- E The curvature of left invariant metrics
- F Comments concerning the Einstein–Boltzmann system
- References
- Index
- [UNTITLED]