TheoremStatement: AI-adaptedProof: Literature-sourced

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Nomizu extension and uniqueness in the analytic Lorentzian setting

Statement

Let $(U, g)$ be a connected simply connected real-analytic Lorentzian manifold. Here $U$ is a smooth $n$ -manifold without boundary, $n \geq 1$ , with a compatible real-analytic atlas, and $g$ is a smooth symmetric nondegenerate covariant two-tensor whose components in that atlas are real analytic. Lorentzian means that at each point some tangent-space basis gives the metric matrix $diag (- 1, 1, \dots, 1)$ , with $n - 1$ positive entries. A smooth vector field $Y$ on an open set is Killing when $L_{Y} g = {\frac{d}{d t} ∣}_{t = 0} Φ_{t}^{*} g = 0$ , where $Φ_{t}$ is its local flow. No orientation or time orientation is assumed.

Every smooth Killing field on a nonempty connected open subset $O$ extends to a unique smooth Killing field on all of $U$ . In particular, two global Killing fields agreeing on a nonempty open subset agree everywhere: restrict to a connected coordinate subneighborhood of that subset and use uniqueness.

Statement references

Smooth manifolds and their smooth charts

An arbitrary metric isometry transports analyticity and Killing fields

Simply connected topological spaces

Facts & Assumptions

[F1]

Smooth manifolds, smooth structures, charts and coordinate domains have the general-dimensional convention in the smooth-manifold definition. Smooth manifolds and their smooth charts

[F2]

Compatible real-analytic atlases, analytic metric components, symmetric nondegenerate metrics and Killing fields have the dimension-independent conventions explicitly formulated in the analytic-pullback supplier. Only these formulation conventions are used here, without imposing the isometry hypotheses of its transport conclusion. The Lorentzian signature is specified inline above. An arbitrary metric isometry transports analyticity and Killing fields

[F3]

Simple connectedness has the nonempty path-connected convention in the cited definition. Simply connected topological spaces

[F4]

The imported Nomizu existence and uniqueness inputs apply to these analytic pseudo-Riemannian data and hence to the Lorentzian subclass; they require neither a field equation nor completeness of the Killing field.

External input

Chruściel, Section 2, Theorem 2.1 and footnote 6, supplies global existence in the analytic pseudo-Riemannian setting. Ionescu–Klainerman, Section 1, p. 2, explicitly supplies uniqueness under the same hypotheses. These established results are imported at statement level; no proof reconstruction is claimed.

Depends on

Used by · 1

A commuting-extension obstruction excludes arbitrary analytic-model isometries

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

P. T. Chruściel, On rigidity of analytic black holes · Section 2, Theorem 2.1 (Nomizu) and footnote 6, printed/PDF p. 6.

A Killing field on a connected open subset of a connected simply connected analytic pseudo-Riemannian manifold extends to the whole manifold.

Hypotheses: An analytic pseudo-Riemannian metric, connected simply connected ambient manifold, and a smooth Killing field on a nonempty connected open set; no field equation or completeness assumption.

Reading limits: Read the web-extracted PDF text of Theorem 2.1 and footnote 6, which explicitly covers pseudo-Riemannian metrics. Existence is imported at statement level; neither Nomizu’s original proof nor its pseudo-Riemannian adaptation was reconstructed.

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.

The preceding extension is unique for a real-analytic pseudo-Riemannian metric when the ambient manifold and initial open set are connected and the ambient manifold is simply connected.

Hypotheses: A smooth Killing field on the connected open set in the real-analytic pseudo-Riemannian manifold; connectedness and simple connectedness as stated. No field equation or boundary condition is assumed for this analytic theorem.

Reading limits: Read the web-extracted PDF text of the opening Section 1 paragraphs and footnote 1. This verifies the statement and its general pseudo-Riemannian setting, not Nomizu’s proof or the later Ricci-flat extension results.

Sources

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.