Relative stationary deformation and extension obstruction

Work in ZFC, including the axiom of choice. A smooth manifold carries a maximal smooth atlas on a Hausdorff second-countable topological manifold. A smooth Lorentzian metric on a smooth four-manifold without boundary is a smooth symmetric nondegenerate covariant two-tensor of signature $(-,+,+,+)$. For a smooth vector field $X$ and a smooth metric $h$, write ${\mathcal{L}}_{X}h={\frac{d}{dt}\mid }_{t=0}{\mathrm{\Phi }}_t^{\ast }h$, where ${\mathrm{\Phi }}_t$ is the local flow of $X$; call $X$ an $h$-Killing field when ${\mathcal{L}}_{X}h=0$. These definitions require no chosen orientation or time orientation. Brackets use $[X,Y]u=X(Yu)-Y(Xu)$ for smooth functions $u$; simply connected means nonempty and path-connected with trivial fundamental group at every basepoint.

A relative stationary deformation datum consists of a connected simply connected smooth four-dimensional manifold $U$ without boundary, a nonempty connected open subset $O\subset U$, smooth Lorentzian metrics $g_0,\tilde{g}$ with $g_0=\tilde{g}$ on $O$, a smooth $\tilde{g}$-Killing field $T$ on $U$, and a smooth $g_0$-Killing field $Z$ on $O$ satisfying $[T{\mathrm{\mid }}_O,Z]=0$ there. The open set $O$ carries the subspace topology and canonical restricted smooth structure of $U$. The matching subset $O$ is part of the datum; no agreement of the two metrics elsewhere is imposed.

The datum has a commuting-extension obstruction when no smooth field $Y$ on all of $U$ obeys ${\mathcal{L}}_Y\tilde{g}=0$, $[T,Y]=0$ and $Y{\mathrm{\mid }}_O=Z$. This is a relative extension problem for a specified germ on a connected matching region, not a claim that no other local Killing field exists.

For a specified smooth Lorentzian four-manifold $(V,g_V)$, with $V$ allowed to be an open neighborhood in a model manifold, an isometry onto the model neighborhood means a smooth diffeomorphism $F:U\to V$ with $F^{\ast }{g}_V=\tilde{g}$. Here a diffeomorphism is a bijective smooth map with smooth inverse, and $(F^{\ast }{g}_V{)}_p(v,w)=(g_V{)}_{F(p)}(d{F}_{p}v,d{F}_{p}w)$ for $v,w\in T_{p}U$. The pushed-forward field is $(F_{\ast }T{)}_{F(p)}=d{F}_p(T_p)$; no condition on $F_{\ast }T$ is part of this definition.

The Axiom of Choice

Smooth manifolds and their smooth charts

An open subset of a smooth manifold has a canonical restricted smooth structure

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

Given: The displayed metrics, open sets, fields and specified model $(V,g_V)$, under the ambient ZFC convention.

Goal: Specify the relative extension problem and model-isometry condition precisely.

1.1 The given smooth structure on $U$ restricts canonically to the open subset $O$ with its subspace topology, making $O$ a smooth four-manifold without boundary. Both metrics are Lorentzian by the pointwise metric definition, without any chosen orientation. The smooth fields on $U$ and on its open submanifold $O$ have local flows. Thus the pullback derivative defines their metric Lie derivatives and the stated Killing predicates on the respective domains. Countable choice needed for the smooth tangent-bundle conventions is available by restricting the ambient axiom of choice to countable families. [A1, F1, F2, F5]

2.1 At each point of $O$, sufficiently short segments of the local flow of $Z$ stay in $O$. Since the two metrics coincide throughout $O$, their pullbacks along those segments coincide, hence so do their derivatives at parameter zero. The Killing equation for $Z$ is therefore the same on $O$ for either metric. Restrictions of fields and their commutator are defined there, using only their action on smooth functions on that open set. [F2, F3, step 1.1]

3.1 The three extension equations are predicates on a smooth field on all of $U$; their joint nonexistence defines a property of the datum without asserting existence of obstructed data. With the model manifold and its metric specified, the tensor pullback and vector pushforward have precisely the displayed meanings. The isometry predicate concerns the metric alone and does not identify the stationary generator with a preferred model field. [A1, F2, F3, F4, step 2.1] ∎