Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

A 1-plane field on a manifold is a smooth choice of a 1 dimensional subspace of the tangent space at every point. If this plane field is integrable then there is an associated 1-dimensional foliation on the manifold that has the plane field as its tangent space. Parameterizing the leaves of the foliation would seem to give a one parameter group of diffeomorphisms. Likewise, to every one parameter group of diffeomorphisms there is an associated vector field (which could be thought of as a smooth 1-plane field), and at any point flowing as long as one could in both directions would seem to give a leaf. The union of these leaves would then seem to give a foliation.

Can someone confirm if this line of reasoning is essentially correct. These ideas seem to be the same except for foliations don't come with a parameterization. I'm guessing there is a natural way to parameterize using the charts though.

share|improve this question

1 Answer 1

up vote 0 down vote accepted

The leaves may not be coherently orientable. So your plane field (really a line field) may not lift to a vector field, let alone a 1-parameter family of diffeomorphisms. You can construct non-orientable line fields in $\mathbb R^2 \setminus \{0\}$.

Provided your line field has no orientability obstruction, it lifts to a vector field. Vector fields don't always integrate to 1-parameter families of diffeomorphisms for completeness issues -- but in various circumstances like if your manifold was compact, then it would have to integrate.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.