LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A Riemannian Killing shift vanishing on the whole boundary vanishes globally

Statement

Work in ZFC. Let $(P, h)$ be a connected smooth Riemannian manifold with nonempty smooth boundary. Let $X$ be a smooth vector field up to the boundary, Killing on the interior in the sense that $L_{X} h = 0$ there, with $X ∣_{\partial P} = 0$ . Then $X = 0$ everywhere on $P$ .

Consequently, let a supplied smooth stationary maximal spacelike hypersurface $\overline{Σ}$ have induced Riemannian metric smooth and positive definite up to its smooth boundary, with its normal and stationary lapse–shift decomposition smooth there. Require that every connected component of $\overline{Σ}$ have nonempty boundary. If $k = 0$ and $T ∣_{\partial \overline{Σ}} = 0$ , then its stationary shift vanishes everywhere. Here stationarity means $L_{T} g = 0$ , and the conventions are $g (n, n) = - 1$ , $N = - g (T, n)$ , $T = N n + X$ and $k (U, V) = - g (\nabla_{U} n, V)$ .

Statement references

Riemannian metric and riemannian manifold

Smooth charts, atlases, and structures with boundary

Fundamental theorem of riemannian geometry

Christoffel formula for the levi civita connection

The Lie derivative of a tensor field

The coordinate formula for the Lie derivative of a covariant tensor

Clairaut--Schwarz theorem for continuous second partial derivatives

Picard-Lindelöf local existence and uniqueness for first-order systems

Stationary Lorentzian data and tensor conventions

Spacelike sections, maximality and lapse–shift data

The Axiom of Choice

Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets

Facts & Assumptions

[A1]

ZFC, including the Axiom of Choice, is the inherited convention for the stationary data. In the stationary application it supplies the countable choice required by the ambient connection interface; the local coordinate propagation below makes no additional family choice. The Axiom of Choice Stationary Lorentzian data and tensor conventions

[F1]

A Riemannian metric is a smooth positive-definite symmetric covariant two-tensor. Boundary charts have relatively open half-space images, with smoothness in the local-extension sense. Thus after shrinking a chart, its finitely many metric and field components have smooth extensions to an open Euclidean neighborhood. Riemannian metric and riemannian manifold Smooth charts, atlases, and structures with boundary

[F2]

A smooth Riemannian metric, including at a smooth boundary, has a unique torsion-free metric-compatible connection $D$ . Its coordinate symbols are $Γ^{ℓ}_{i j} = \frac{1}{2} h^{ℓ a} (\partial_{i} h_{j a} + \partial_{j} h_{i a} - \partial_{a} h_{i j})$ . Fundamental theorem of riemannian geometry Christoffel formula for the levi civita connection

[F3]

For a smooth vector field $V$ with local flow $Φ_{t}$ and a smooth covariant two-tensor $m$ , the Lie derivative is $L_{V} m = {\frac{d}{d t} ∣}_{t = 0} Φ_{t}^{*} m$ on each local flow domain. In coordinates, $(L_{V} m)_{i j} = V^{a} \partial_{a} m_{i j} + m_{a j} \partial_{i} V^{a} + m_{i a} \partial_{j} V^{a}$ . This is used on boundaryless interiors and ambient charts; smooth identities then extend to boundary points by continuity. The coordinate formula for the Lie derivative of a covariant tensor The Lie derivative of a tensor field

[F4]

Second partial derivatives of a $C^{2}$ function on an open Euclidean set commute. At boundary points this applies to the local smooth extensions in [F1]. Clairaut--Schwarz theorem for continuous second partial derivatives

[F5]

For a continuous first-order system on an open time–state domain, locally Lipschitz in the state, two solutions with the same initial data agree on every common subinterval containing the initial time. In particular this applies to $z^{'} = B (t) z$ when the matrix $B$ is smooth. Picard-Lindelöf local existence and uniqueness for first-order systems

[F6]

The stationary data supply the smooth Lorentzian metric, its torsion-free metric-compatible connection $\nabla$ , and $L_{T} g = 0$ . The hypersurface conventions supply the positive-definite induced metric, smooth unit normal, $N = - g (T, n)$ , $X = T - N n$ , and $k (U, V) = - g (\nabla_{U} n, V)$ . Stationary Lorentzian data and tensor conventions Spacelike sections, maximality and lapse–shift data

[F7]

A topological space is connected when it has no separation into two nonempty disjoint open subsets. Equivalently, its only clopen subsets are the empty set and the whole space. Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets

Proof

Given: The smooth boundary and Killing data of the Statement, with [A1] and [F1]–[F7].

Goal: Propagate the zero first jet from a boundary point and apply the result to the stationary shift.

1.1givenF1F2F3

For any smooth metric $m$ with torsion-free compatible connection $\nabla^{m}$ , expanding [F3] and using compatibility gives $(L_{V} m) (U, W) = m (\nabla_{U}^{m} V, W) + m (U, \nabla_{W}^{m} V)$ . Apply this to $h$ on the interior and extend to the boundary by smoothness. Thus $A_{p} (v) = D_{v} X$ is $h$ -skew at every $p$ . At $p \in \partial P$ , differentiating the zero restriction of $X$ in each boundary tangent direction gives $A_{p} (v) = 0$ for $v \in T_{p} \partial P$ . If $ν$ is a unit normal to that hyperplane in $T_{p} P$ , skewness gives $h (A_{p} ν, v) = - h (ν, A_{p} v) = 0$ for all boundary tangent $v$ , and $h (A_{p} ν, ν) = 0$ . Positive definiteness implies $A_{p} ν = 0$ , so $(X, D X)_{p} = (0, 0)$ . This also covers dimension one, where the tangent hyperplane is zero dimensional.

1.2F1F2F3F4step 1.1algebra

In a coordinate chart put $u_{j} = h_{j a} X^{a}$ and $E_{i j} = 2 Γ^{ℓ}_{i j} u_{ℓ}$ . The coordinate formula in [F2] and the Lie derivative formula in [F3] give the Killing equation $\partial_{i} u_{j} + \partial_{j} u_{i} = E_{i j}$ . Differentiate its $(j, k)$ equation by $i$ , its $(i, k)$ equation by $j$ , and subtract the derivative by $k$ of its $(i, j)$ equation. By [F4] the left side is $2 \partial_{i} \partial_{j} u_{k}$ . Consequently $\partial_{i} \partial_{j} u_{k} = (\partial_{i} Γ^{ℓ}_{j k} + \partial_{j} Γ^{ℓ}_{i k} - \partial_{k} Γ^{ℓ}_{i j}) u_{ℓ} + Γ^{ℓ}_{j k} \partial_{i} u_{ℓ} + Γ^{ℓ}_{i k} \partial_{j} u_{ℓ} - Γ^{ℓ}_{i j} \partial_{k} u_{ℓ} .$ The equality holds up to the boundary by continuity. Hence the finite list $Y = (u_{j}, \partial_{i} u_{j})$ satisfies $\partial_{r} Y = A_{r} (x) Y$ for smooth matrix functions $A_{r}$ : the derivatives of its first entries are entries of $Y$ , and the displayed identity supplies the derivatives of all remaining entries. These matrices depend only on $h$ and its derivatives.

2.1givenF1F2F5step 1.2

Let $p$ have $(X, D X)_{p} = (0, 0)$ . Choose a sufficiently small convex coordinate ball, or ball intersected with the closed half-space at a boundary point. By [F1] the metric coefficients extend smoothly to a full ball about $p$ ; after shrinking, positive definiteness persists, so the Christoffel formula extends each matrix $A_{r}$ smoothly too. For any $q$ in a smaller such relative ball, the straight coordinate segment $x (t) = x (p) + t (x (q) - x (p))$ , $0 \leq t \leq 1$ , lies in the chart, and $Y (x (t))$ satisfies $z^{'} = B (t) z$ , where $B (t) = (x (q) - x (p))^{r} A_{r} (x (t))$ extends smoothly to an open interval containing $[0, 1]$ . Since $X_{p} = 0$ , differentiation of $u_{j} = h_{j a} X^{a}$ gives $\partial_{i} u_{j} (p) = h_{j a} (D_{i} X)^{a} (p) = 0$ , hence $z (0) = 0$ . To apply [F5] with an interior initial time even if $p$ is on the boundary, extend this solution by zero for small negative $t$ : the right derivative at zero is $B (0) z (0) = 0$ , so this is a $C^{1}$ solution across zero. It agrees with the zero solution on $[0, 1]$ by [F5]. Thus $Y (q) = 0$ throughout the smaller relative neighborhood; equivalently $X = D X = 0$ there.

3.1givenF1F2F7step 1.1step 2.1

The set $Z = {p \in P : X_{p} = 0, (D X)_{p} = 0}$ is closed by smoothness and open by step 2.1. It is nonempty by step 1.1 and the nonempty boundary. Connectedness of $P$ gives $Z = P$ , proving the first assertion. This argument uses relative neighborhoods at the boundary and does not require a flow or boundary normal geodesics.

4.1givenA1F1F2F3F6step 1.1step 3.1∎

For the stationary application, define ${\hat{D}}_{U} V = (\nabla_{U} V)^{⊤}$ for tangent fields on the hypersurface. The ambient torsion-free property and tangency of the bracket give zero torsion, and ambient metric compatibility gives $U h (V, W) = h ({\hat{D}}_{U} V, W) + h (V, {\hat{D}}_{U} W)$ . By uniqueness in [F2], $\hat{D} = D$ for the induced metric. The identity proved in step 1.1, now for $g, T$ , gives $0 = g (\nabla_{U} T, V) + g (U, \nabla_{V} T)$ . Substitute $T = N n + X$ . Orthogonality removes the derivatives of $N$ , and the definition of $k$ yields $h (D_{U} X, V) + h (U, D_{V} X) = N (k (U, V) + k (V, U))$ . If $k = 0$ this is the Riemannian Killing equation. At the boundary $T = 0$ implies $N = - g (T, n) = 0$ and then $X = T - N n = 0$ . Apply step 3.1 separately to each connected component, which by hypothesis has nonempty boundary. Therefore $X = 0$ everywhere. This deduction uses no vacuum equation and no null-sheet hypothesis.

Depends on

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.