# Integrability of Continuous Tangent Subbundles

Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space?

More specifically given a smooth manifold of $M$ and a continuous sub-bundle $E$ of $TM$, by integrating I mean finding a foliation whose leaves have tangent spaces that coincide with $E$ (this question also locally happens to be equivalent to solving a 1st order system of linear homogeneous PDE). In dynamical systems such sub-bundles arise as sub-spaces that are left invariant by differential of some diffeomorphism and sometimes being able to integrate them helps for certain classifications or statistical properties.

I am wondering if continuous sub-bundles OR continuous linear homogeneous systems of PDEs appear else where where it is important to know whether if you can integrate them or not?

Thanks

• Every time anyone applies the Frobenius theorem they are doing exactly that. – Igor Khavkine May 20 '15 at 6:46
• Standard Frobenius requires at least some bit of differentiability. So if you only have continuity, what do you expect to get? What kind of submanifolds do you expect to get? – Stefan Waldmann May 20 '15 at 7:55
• @igor you can't apply frobenius directly when the subbundle is not C^1 please read the question more carefully. – Avicenna May 20 '15 at 10:28
• A 1-dim continuous sub bundle is generated by a non trivial continuous vector field, right? Then often control theory, transport theory, fluid theory deal with integrating such vector fields. Is this all under "dynamical systems"? Smoothness along the leaves paired with global continuity appears, for example, in some problems in Poisson geometry like, for example, this one: projecteuclid.org/euclid.ajm/1355321986 – Nicola Ciccoli May 21 '15 at 17:54
• Another field of math where this kind of things may happen is geometric quantization where it is not unlikely to find Lagrangian polarizations which are densely smooth and only continuous on singularities (tipically with square-root like isolated singularities) – Nicola Ciccoli May 21 '15 at 18:02