Alphabeta MathSupporting prerequisites · 01 — Gauss lemma
Presents 20 supporting prerequisites, covering published library prerequisites and dependency levels 0–2.
1 · Prerequisites
- Fundamental theorem of riemannian geometry
- The exponential map scales geodesic time
- Geodesics have constant speed for a metric-compatible connection
- Levi civita connection
- Metric compatible connection on a riemannian vector bundle
- Torsion free is equivalent to symmetric christoffel symbols in coordinate frames
- A function continuous on an interval I whose derivative vanishes at every interior point of I is constant on I; consequently two such functions with the same derivative differ by a constant
- The Axiom of Countable Choice (AC_ω)
- Stokes theorem for the standard simplex
- Smooth singular chain and cochain complexes
- Integral of a form over a smooth singular simplex
- Simplex integrals are independent of affine coordinate identification
- The exterior derivative commutes with pullback
- Pullback of forms is smooth functorial and preserves wedges
- Domain and exponential map of a connection
- Existence uniqueness and smooth dependence of geodesics
- Adopted spacetime model and classical domain of the gravitational framework
- Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions
- Stress-energy tensor field of a model
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- The scalar-field matter model of the classical gravitational framework
- Metric compatibility, torsion freedom and the metric-selected connection class
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- SI unit and coordinate conventions of the classical gravitational framework
- Cʳ and smooth maps between smooth manifolds
- The Axiom of Choice
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- A smooth vector field is a smooth section of the tangent bundle
- The fundamental theorem on flows
- The Lie derivative of a tensor field
- The Lie bracket of smooth vector fields
- Simply connected topological spaces
- Diffeomorphisms and local diffeomorphisms of manifolds
- The pullback of a covariant tensor field
- Pushforwards and pullbacks of vector fields by a diffeomorphism
- Smooth manifolds and their smooth charts
- An open subset of a smooth manifold has a canonical restricted smooth structure
- Smooth charts, atlases, and structures with boundary
- Smooth maps between manifolds with boundary
- Riemannian metric and riemannian manifold
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- Through each point there is a unique maximal integral curve
- Local and global flows generated by a vector field
- Riemannian volume is the radon measure of the riemannian density
- Coordinate formula for riemannian divergence
- The determinant differential is D det(A)[H]=tr(adj(A)H) at every matrix, and Jacobi's formula holds on the invertible locus
- A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions
- Embedded smooth submanifolds with boundary
- Affine connection on a smooth manifold
- Torsion tensor of an affine connection
- The coordinate formula for the Lie derivative of a covariant tensor
- Coordinate formula for the Lie bracket
- Gradient hessian and divergence connection formulas
- Laplace–Beltrami operator as the trace of the Hessian
- Riemannian gradient
- The riemannian volume density is coordinate independent
- Positive-part truncation calculus and admissible cut-off weak tests
- Smooth partitions of unity exist on manifolds
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Collar neighborhood theorem
- Riemannian distance on a connected manifold
- The riemannian distance topology is the manifold topology
- The differential of a smooth real-valued function
- The standard smooth step function
- The chain rule for differentials of smooth maps
- Riemannian speed and length
- Continuous functions have an oriented Riemann integral
- The oriented Riemann integral is linear in continuous integrands
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- Smooth collars of a manifold boundary
- Real analytic germs in several variables
- Smooth atlases
- A smooth tensor field
- Pullback of covariant tensors is smooth and functorial
- Diffeomorphism pushforward preserves Lie brackets
- Piotr T. Chruściel, Geometry of Black Holes (revised July 2018; August 15, 2018 PDF) · Appendix A.5, printed p. 233 (PDF p. 241), symmetric/nondegenerate two-tensors and signature; Appendix A.21, printed p. 305 (PDF p. 313), equation (A.21.1).
- Causal curves, chronological and causal reachability, and inextendibility
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- Spacelike sections, maximality and lapse–shift data
- Killing fields of a Lorentzian metric
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- 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
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- AC implies DC implies countable choice
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- Products of smooth manifolds have a canonical product smooth structure
- The timelike set has exactly two convex components, described in orthonormal coordinates
- P. T. Chruściel, Elements of causality theory · §§2.1, 2.3–2.4; Definition 2.1.1, printed p. 4; curve convention p. 16; futures/pasts p. 17; Proposition 2.4.2(3), p. 18; Corollary 2.4.11 and Proposition 2.4.12, pp. 24–25 (PDF pp. 7, 19–21, 27–28).
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · §1.2, p. 6 (dimension convention); §2.3, Remark 2.8, p. 28 (PDF pp. 6, 28).
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 — Lorentzian geometry and time orientation · §1.2, printed p. 6 (dimension and signature); §1.3, printed p. 7 (Lorentz scalar product, timelike cones and choice of future); §1.5, printed p. 9 (Lorentzian manifold); §1.7, printed pp. 10–11 (continuous time orientation). PDF pp. 6–7, 9–11.
- Cᵏ maps and multi-index derivative notation in Euclidean space
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Sum, product, quotient, and chain rules from difference quotients
- Clairaut--Schwarz theorem for continuous second partial derivatives
- Tensor transition laws define a smooth vector bundle
- Smoothness of a tensor field is equivalent to smooth coordinate components
- The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
- A closed subset of a compact metric space is compact
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- For n ≥ 1 every bounded sequence in ℝⁿ has a convergent subsequence
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- A compact subset of a metric space is closed and bounded
- Distance from a point to a subset
- Distance to a nonempty subset is one lipschitz
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Paths, path-connected spaces and path components
- For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected
- Every path-connected space is connected, and every path component lies inside a component
- Connected components, quasicomponents, and totally disconnected spaces
- Open cover, subcover, compact metric space, and compact subset of a metric space
- A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- Interior and boundary of a manifold with boundary
- Interior, closure, boundary, limit point, isolated point and dense subset of a metric space
- Christoffel formula for the levi civita connection
- Picard-Lindelöf local existence and uniqueness for first-order systems
- Stationary Lorentzian data and tensor conventions
- Differentiated asymptotic bounds and parameter uniformity
- Chronology, causality, strong causality, global hyperbolicity and time functions
- A vector field is complete if and only if its flow is global
- The smooth inverse function theorem on manifolds
- Embedded submanifolds and slice charts
- The inclusion of an embedded submanifold is a smooth embedding
- Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form
- Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor
- Product connection on tensor and hom bundles
- Induced connections commute with contraction and permutation
- Sine and cosine defined by their real power series
- The sine and cosine power series converge absolutely for every real argument
- The sum of a real power series is real analytic throughout the open interval determined by its radius
- Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero
- Parity and the Pythagorean identity for sine and cosine
- Pi is the first positive zero of sine
- Causal solutions, Killing horizons, escape and metric extensions
- Ionescu–Klainerman, On the local extension of Killing vector-fields in Ricci flat manifolds · §3.1, equations (3.2)–(3.4) and the region R immediately after (3.4), printed/PDF pp. 15–16
- P. T. Chruściel, On rigidity of analytic black holes · Section 2, Theorem 2.1 (Nomizu) and footnote 6, printed/PDF p. 6.
- Ionescu–Klainerman, On the local extension of Killing vector-fields in Ricci flat manifolds · Section 1, printed/PDF p. 2, opening extension-problem paragraph and footnote 1.
- Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- The Borel sigma-algebra of a topological space
- Sigma-algebras
- Measures on sigma-algebras
- Measure-preserving transformations and systems
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- A point lies in the closure of A iff every basic neighbourhood of it meets A; the closure is the smallest closed superset and equals A together with its derived set
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
Theorem
Gauss lemma
Theorem
Stokes theorem for smooth singular chains
Theorem
The exponential domain is open and the exponential map is smooth
Postulate
Adopted Einstein field equation in the direct-equation presentation
Definition
Relative stationary deformation and extension obstruction
Lemma
Complete volume preserving shift flow on a spherical exterior
Lemma
Maximal stationary shift has zero divergence and tangent boundary
Lemma
Weak compact-domain comparison for continuous nonnegative potentials
Lemma
The radius of a graph section is bounded on a compact boundary collar
Lemma
An arbitrary metric isometry transports analyticity and Killing fields
Lemma
Inaccessible set is the closure of its relative interior
Lemma
All-jet reflection parity from an exact local boost
Lemma
A smooth graph section with a genuine end has a compact inside core
Lemma
A Riemannian Killing shift vanishing on the whole boundary vanishes globally
Lemma
Orthogonal flow coverage gives a global static product and the needed end bounds
Lemma
The stationary maximal vacuum divergence identity with the article sign
Lemma
The bracket of two Killing fields is Killing
Lemma
Analytic Kerr coordinates across the chosen nondegenerate horizon
Theorem
Nomizu extension and uniqueness in the analytic Lorentzian setting
Lemma