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A commuting-extension obstruction excludes arbitrary analytic-model isometries
Statement
A relative stationary deformation datum with a commuting-extension obstruction as in Relative stationary deformation and extension obstruction admits no compatible real-analytic structure making its deformed metric analytic. Consequently it cannot be isometric, by any smooth diffeomorphism, to an analytic model neighborhood. The isometry need not preserve , or the matching coordinates. This statement concerns an isometry defined on the entire specified ; it does not exclude Kerr germs at particular points in a part where the deformation vanishes.
Facts & Assumptions
The relative datum has connected simply connected , connected nonempty , and prohibits a commuting extension. Relative stationary deformation and extension obstruction
For an analytic metric Nomizu supplies a global Killing extension and uniqueness on nonempty open sets. Nomizu extension and uniqueness in the analytic Lorentzian setting
Under this uniqueness property every extension commutes with . Uniqueness forces an extended axial field to commute
An isometry to an analytic metric supplies an analytic pullback structure. An arbitrary metric isometry transports analyticity and Killing fields
Proof
Given: An obstructed relative datum.
Goal: Exclude an analytic presentation and an arbitrary isometry onto an analytic model.
Suppose a compatible real-analytic structure makes analytic. The field is a smooth Killing field on the connected open set , since the metrics agree there. Apply Nomizu with the connected simply connected manifold to obtain a global Killing extension and the required uniqueness property. The commutation lemma yields , contradicting the obstruction.
If an isometry onto an analytic model existed, its pulled-back atlas would supply exactly the analytic structure excluded in step 1.1. Hence no such isometry exists, independently of the images of the original symmetry fields. Neither step asserts that every smaller neighborhood carries an obstruction.
Depends on
Dependency tree · 2 levels
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