lin-0076

5.4 Bracket residuals and integrability

Let \(g\) be a Riemannian metric on \(\mathcal M\), and let \(P_{\mathrm{vis}}:T\mathcal M\to \mathcal D_{\mathrm{vis}}\) be the orthogonal projection on the constant-rank stratum.

Definition 5.5 Visible bracket residual

For \(u,v\in \Gamma (\mathcal D_{\mathrm{vis}})\), define

\[ r(u,v) = (I-P_{\mathrm{vis}})[u,v] \in \Gamma (\mathcal D_{\mathrm{vis}}^\perp ). \]

The projection makes the obstruction measurable. Its magnitude depends on the metric, but its vanishing does not.

Proposition 5.6 Residual criterion

For a local frame \(w_1,\ldots ,w_r\) of a constant-rank distribution, the following are equivalent:

  1. \(r(w_a,w_b)=0\) for every \(a,b\);

  2. \([w_a,w_b]\in \Gamma (\mathcal D_{\mathrm{vis}})\) for every \(a,b\);

  3. \(\mathcal D_{\mathrm{vis}}\) is involutive.

Proof ▼

The first two statements are equivalent by projection. If the frame brackets lie in the distribution, the Leibniz rule shows that the bracket of any two smooth linear combinations of the frame does also. The converse follows by applying involutivity to the frame sections.

Corollary 5.7 Frobenius integrability

On a constant-rank stratum, every visible bracket residual vanishes if and only if \(\mathcal D_{\mathrm{vis}}\) is locally tangent to a foliation.

Proof ▼

Apply the classical Frobenius theorem [ Lee , 2012 ] to the residual criterion.

This is a closure theorem. It is not an acyclicity test, a causal-sufficiency criterion, or an identification theorem.