Alphabeta MathStationary geometry and analytic data
Define the common Lorentzian, causal, horizon, hypersurface and field conventions, together with weighted elliptic, graph, continuation and area data. These fifteen mathematical interfaces supply both later prerequisite pages without importing article-specific conclusions.
1 · Prerequisites
- The Borel sigma-algebra of a topological space
- The Euclidean implicit function theorem with derivative formula
- The Axiom of Choice
- Smooth manifolds and their smooth charts
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
- Products of smooth manifolds have a canonical product smooth structure
- Cʳ and smooth maps between smooth manifolds
- Regular and critical points and values
- Codimension and hypersurfaces
- Embedded submanifolds and slice charts
- A regular level set is an embedded submanifold
- Slice-chart restrictions form a smooth atlas
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Second countability: an at most countable basis for the topology
- Assuming countable choice, every second countable space is Lindelöf
- The mean value inequality: if f : [a,b] → ℝᵐ is continuous and differentiable on (a,b) with ‖ f'‖₂ ≤ M, then ‖ f(b)-f(a)‖₂ ≤ M(b-a)
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Fundamental theorem of riemannian geometry
- Domain and exponential map of a connection
- Existence uniqueness and smooth dependence of geodesics
- The smooth inverse function theorem on manifolds
- The Axiom of Countable Choice (AC_ω)
- Smooth embeddings
- The image of a smooth embedding is an embedded submanifold
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Metric compatibility, torsion freedom and the metric-selected connection class
- Diffeomorphisms and local diffeomorphisms of manifolds
- Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ
- Topological manifolds with and without boundary
- Killing fields of a Lorentzian metric
- Causal curves, chronological and causal reachability, and inextendibility
- Chronology, causality, strong causality, global hyperbolicity and time functions
- Local and global flows generated by a vector field
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Section 2.6, Definition 2.16, printed/PDF p. 33; section 4.4, opening cone-order convention and Definition 4.89, printed/PDF p. 137.
- E. Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity, arXiv:gr-qc/0703035v1 · Section 3.2.1, printed p. 39, PDF page 39, opening paragraph.
- Measurable domains on smooth initial surfaces
- Riemannian metric and riemannian manifold
- Lebesgue density theorem
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Coordinate derivations form a basis of the tangent space
- Change-of-coordinate formula for tangent bases
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Lemma
Transverse hitting maps have local differentiable extensions
Lemma
Admissible domains for outward normal graphs
Definition
Asymptotically flat spacelike end and derivative decay
Definition
Ambient closure and intrinsic boundary of a hypersurface
Definition
Causal exterior, chronological boundaries and Cauchy data
Definition