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
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Local and global flows generated by a vector field
- Metric compatibility, torsion freedom and the metric-selected connection class
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- The derivative of the metric density and the chart formula for the divergence of a vector field
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
- The Axiom of Countable Choice (AC_ω)
- Smooth embeddings
- The image of a smooth embedding is an embedded submanifold
- Diffeomorphisms and local diffeomorphisms of manifolds
- E. Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity (2007) · Sections 2.3.1-2.3.2, printed pp. 21-22, equations (2.27)-(2.31); section 2.3.4, printed pp. 23-24, equations (2.36), (2.42); section 2.4.2, printed p. 31, equations (2.74)-(2.77).
- Embedded submanifolds and slice charts
- The timelike set has exactly two convex components, described in orthonormal coordinates
- Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral
- The Axiom of Choice
- The differential of a smooth map
- The pullback of a covariant tensor field
- Coordinate formula for the differential
- The chain rule for differentials of smooth maps
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- The Darboux and Riemann definitions agree: a bounded f on [a,b] is Darboux integrable with integral I if and only if for every real ε > 0 there is a real δ > 0 such that |S(f,P,ξ) - I| < ε for every tagged partition of mesh below δ
- The inner-product norm is definite, homogeneous, and satisfies the triangle inequality
- Continuous functions on a closed interval attain their extrema
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Banach space
- C k map between Banach spaces
- Fredholm operator cokernel and index
- Compact linear operator
- Implicit function theorem for Banach spaces
- Fredholm index is stable under compact perturbations
- The standard product norms on a finite product of normed spaces
- Finite products of Banach spaces are Banach
- In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle
- Fréchet derivative between Banach spaces
- Metric continuity characterisations, with countable choice for the sequential converse
- The reals are complete
- Riemannian metric and riemannian manifold
- The Borel sigma-algebra of a topological space
- Riemannian volume density
- The riemannian volume density is coordinate independent
- Countable partition construction of the Borel set function
- Intrinsic density measure and its chart restriction
- Riemannian volume is the radon measure of the riemannian density
- The completion domain and proposed completed set function of a measure space
- Assuming countable choice, every measure space has a unique complete extension to its completion
- Riemannian distance on a connected manifold
- Riemannian distance is a metric
- The riemannian distance topology is the manifold topology
- Unnormalised Hausdorff measure
- Euclidean Hausdorff measure is proportional to Lebesgue measure
- Hausdorff measure is metric and measures every Borel set
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Lipschitz maps control Hausdorff measure
- Coordinate criterion for a riemannian metric
- Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- Riemannian speed and length
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- Line-integral estimates by arc length and the supremum of the field
- Assuming countable choice, every second countable space is Lindelöf
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
Definition
Normal graphs and the nonlinear mean-curvature map
Definition
Periodic isometries and invariant normal averaging
Definition
Fredholm and Banach continuation interfaces
Definition