A continuous symmetric coefficient orthogonal to every compactly supported variation vanishes

Given: A smooth four-manifold $M$, a smooth symmetric nondegenerate covariant $2$-tensor field $g$, a chart $(U,x)$ with metric density $w>0$, and a continuous symmetric field $F$ on $U$ with the stated orthogonality property.

In each chart $g^{ab}$ is smooth: the adjugate formula writes each entry as a polynomial in the smooth $g_{cd}$ divided by the nowhere-zero determinant (If $\det (A)$ is a unit, then $A^{-1}=\det (A{)}^{-1}\mathrm{adj}(A)$). Matrix multiplication gives $g^{ac}{g}_{cb}={\delta }^a{}_b$. Thus raising $q^{ab}=g^{ac}{g}^{bd}{q}_{cd}$ and lowering $q_{ab}=g_{ac}{g}_{bd}{q}^{cd}$ are inverse operations preserving symmetry, smoothness and support. Also $w$ is smooth and positive, since the determinant is smooth, nowhere zero and has constant sign near each point.

Components prescribed smoothly with compact support inside one chart extend by zero to a smooth field on $M$. For a covariant $2$-tensor this follows from the smooth Jacobian transformation law of Chart independence and the transformation law of a smooth covariant 2-tensor field on overlaps; outside its closed support it is zero on a neighborhood. For vectors and covectors the same assertion follows from the coordinate-basis change $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{y}^a=(\mathrm{\partial }{x}^c\mathrm{/}\mathrm{\partial }{y}^a)\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^c$ and its dual: all transition coefficients are smooth (Smooth manifolds and their smooth charts). Compact support in the chart ensures smoothness at points outside the chart as well.

For $0<r<R$ there is a smooth function $\rho :{\mathbb{R}}^4\to [0,1]$ with $\rho =1$ on the closed ball of radius $r$ about $0$ and support in the open ball of radius $R$; after translating and rescaling the chart coordinates this gives, about every point $p\in U$, nonnegative smooth bumps supported in any prescribed coordinate ball contained in $U$ and equal to $1$ near $p$ (A smooth bump between concentric Euclidean balls).

A continuous compactly supported function on a chart image is extended by zero to ${\mathbb{R}}^4$ and integrated on a nondegenerate closed rectangle containing its support in its interior. The extension is continuous, hence integrable by Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable. The integral does not depend on this rectangle: extending a grid to a larger rectangle adds only zero cells, and a common containing rectangle compares any two choices. If this function is nonnegative and positive at a point, continuity gives a smaller closed rectangle $Q_0$ and $m>0$ on which it is at least $m$. A grid containing the endpoints of $Q_0$ has lower sum at least $m\mathrm{vol}(Q_0)>0$; the integral is at least this sum by The lower and upper Darboux integrals over a nondegenerate rectangle in ${\mathbb{R}}^m$, Lower and upper Darboux sums over a grid partition in ${\mathbb{R}}^m$, and Grid partitions of a rectangle in ${\mathbb{R}}^m$, their cells, refinements and mesh. A negative value is treated by changing sign.