Meaning of integrable hyperplane field

Question

Can any on tell the meaning of integrable hyperplane field geometrically?

Why the condition using 1-form for integrable hyperplane field make sense?

Thanks...

Answer 1

1) A hyperplane field is integrable if for each point in space there exists an integral hypersurface of the field passing through this point.

2) If (and only if) a hyperplane field is integrable, there eixsts a change of coordinates such that, in the new coordinates, the field consists of hyperplanes parallel to the hyperplane ${x_1=0}$. Any 1-form specifying the field in the new coordinates is $\alpha=f(x)dx_1$ (with an arbitrary nonzero differentiable function $f$) and obviously satisfies $\alpha\wedge d\alpha=0$; hence the same is true in the original coordinates. Conversely, if $\alpha\wedge d\alpha=0$, then there exists a nonzero function $g$ such that $d(g\alpha)=0$ and hence locally $g\alpha=dF$ for some function $F$ and the integral surfaces are given by $F=\operatorname{const}$.