Constant combinations preserve horizon symmetries and normalize periods

The Lorentzian metric and time orientation are the geometric data of the Lorentzian-manifold definition, and the orientation restricts from $M$ to $U$. Both fields are Killing for the single restricted metric. Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model Oriented smooth manifolds and oriented charts Killing fields of a Lorentzian metric

On an oriented time-oriented smooth Lorentzian four-manifold, clause (a) of the Killing-characterization theorem gives ${\mathcal{L}}_{X}g=0$ if and only if ${\mathrm{\nabla }}_i{X}_j+{\mathrm{\nabla }}_j{X}_i=0$, where indices are lowered using $g$ and $\mathrm{\nabla }$ is its metric-selected connection. The same characterization applies on the open manifold $U$. Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor

The Lie bracket is the commutator $[X,Y]f=X(Yf)-Y(Xf)$ on smooth functions; this formula gives bilinearity over real constants and skew symmetry. The Lie bracket of smooth vector fields

A local flow has an open domain containing $\{0\}\times U$, interval time fibres, the identity at time zero, integral-curve time maps and the flow law whenever both sides are defined. An integral curve $\gamma$ of $X$ satisfies ${\gamma }^{\mathrm{\prime }}(t)=X_{\gamma (t)}$. The flow is global when its domain is $\mathbb{R}\times U$. Local and global flows generated by a vector field Integral curves of a vector field

The differential chain rule for smooth maps gives $\frac{d}{dt}\gamma (ct)=c{\gamma }^{\mathrm{\prime }}(ct)$ for a smooth curve $\gamma$ and a real constant $c$, on the reparametrized interval. The chain rule for differentials of smooth maps

The least common period convention is that of a normalized periodic action: it concerns the entire global flow, rather than requiring every orbit to have the same least period. Only this period convention is used here, not the normal-averaging construction. Periodic isometries and invariant normal averaging

The tensor-derivative theorem applies to a smooth symmetric nondegenerate metric on a smooth four-manifold without boundary, assuming ${\mathrm{A}\mathrm{C}}_{\omega }$ for the canonical tangent-bundle smooth structure. It applies to $g{\mathrm{\mid }}_U$ because the restricted Lorentzian metric is smooth and nondegenerate and $U$ is an open four-manifold without boundary. Clause (c), specialized to a smooth covector $\omega$, gives ${\mathrm{\nabla }}_i{\omega }_j={\mathrm{\partial }}_i{\omega }_j-{\sum }_d{\mathrm{\Gamma }}^d{}_{ji}{\omega }_d$, where $d$ runs over the four chart indices. This formula is linear in $\omega$ over real constants; its finite expansion uses no additional choice. Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields The Axiom of Countable Choice (AC_ω) Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Under the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ convention, a smooth vector field on $U$ is a smooth section of $TU\to U$. On each smooth chart, a vector field is smooth exactly when its coordinate components are smooth functions; this tests smoothness of the section in the induced tangent-bundle charts. A smooth vector field is a smooth section of the tangent bundle Smoothness of a vector field is equivalent to smooth coordinate components