Alphabeta MathHarrison fields, horizons and uniqueness inputs
Connect the common data to Ernst potentials, Harrison transformations, invariant horizon tensors, stationary slices and established uniqueness criteria. Keep each imported result tied to its stated hypotheses and each local consequence tied to its existing argument before article applications.
1 · Prerequisites
- The Axiom of Choice
- Closed and exact differential forms
- De rham cohomology
- The pullback of a differential form
- The exterior derivative commutes with pullback
- De rham cohomology of spheres
- De rham cohomology is smooth homotopy invariant
- Finite chart localization of compactly supported top forms, including boundary
- Nonzero total integral obstructs exactness on a closed manifold
- Smooth manifolds and their smooth charts
- Diffeomorphisms and local diffeomorphisms of manifolds
- Products of smooth manifolds have a canonical product smooth structure
- The n-sphere with its standard smooth atlas
- Oriented smooth manifolds and oriented charts
- Embedded submanifolds and slice charts
- A smooth differential k-form
- The wedge product of differential forms
- Normal and conormal bundles of an embedded submanifold
- Assuming countable choice, normal and conormal bundles are smooth vector bundles
- Vector bundle charts and transition functions
- Pullback of forms is smooth functorial and preserves wedges
- Cartan's magic formula
- The Lie derivative of a tensor field
- Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral
- The local coordinate formula for the exterior derivative
- The fundamental theorem of calculus and integration by parts
- Local coordinate expression for a differential form
- Interior product of a form by a vector field
- The Lie derivative of a differential form
- The exterior derivative of a function is its differential
- Interior product is a graded antiderivation
- Differential forms form a graded commutative algebra
- The exterior derivative is a graded derivation
- The exterior derivative squares to zero
- Complex self-dual forms and metric-dependent contraction
- For A∈ Mₙ(R) and columns u,v over a commutative ring, det(A+uv^T)=det(A)+v^Tadj(A)u
- Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique
- Lorentzian exterior calculus and Hodge star
- Conformal Hodge scaling in every degree
- Orientation of a finite-dimensional real vector space
- A real linear isomorphism preserves or reverses orientation according to the sign of its determinant
- Linear pullback respects tensor products and permutations
- The wedge product of alternating covectors
- Bifurcate Killing horizons, rotation and normalization
- Periodic isometries and invariant normal averaging
- Metric compatibility, torsion freedom and the metric-selected connection class
- Local and global flows generated by a vector field
- Through each point there is a unique maximal integral curve
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
Periods and primitives on a spherical exterior
Lemma
Relative homotopy primitive on a normal bundle
Lemma
Stationary lifting of a spatial primitive
Lemma
Rank-one tangent changes, signature and Hodge transfer
Lemma