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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

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  • $\begingroup$ Every time anyone applies the Frobenius theorem they are doing exactly that. $\endgroup$ May 20, 2015 at 6:46
  • $\begingroup$ 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? $\endgroup$ May 20, 2015 at 7:55
  • $\begingroup$ @igor you can't apply frobenius directly when the subbundle is not C^1 please read the question more carefully. $\endgroup$
    – Avicenna
    May 20, 2015 at 10:28
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    $\begingroup$ 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 $\endgroup$ May 21, 2015 at 17:54
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    $\begingroup$ 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) $\endgroup$ May 21, 2015 at 18:02

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