The title is maybe a little misleading, but here we go.

Not so much a standard problem, but a neat little result. Let . Let be a nowhere vanishing vector field on U. Then for each point there is a surface passing through such that is normal to the surface if and only if .

This is nice since it ties back to the Frobenius posts. If in coordinates, then define . Some texts use “musical notation,” which is amazingly effective once you get used to it. In which case, we would just say let . Now is a smooth 1-form, so it defines a 2-dimensional distribution on in the standard way .

Now by definition, , so our problem has been rephrased in Frobenius language to say D is integrable iff since an integral manifold for through p will satisfy the surface condition.

Thus by the Frobenius Theorem, the problem is reduced to showing is involutive iff . But D is involutive iff given any two smooth sections Y and Z of D, . In fancy notation, , in layman’s terms, if , then (maybe a sign is wrong there, it doesn’t really matter to finish the problem, but I didn’t actually work it out so don’t fully trust it).

So we now have D involutive iff . But this happens iff iff has trivial projection onto , which is precisely .

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