Let be $ M$ a $ n-$ dimensional manifold. A distribution $ D$ on $ M$ is an assignment of subspace $ D_m \subset T_mM$ , for all $ m\in M$ .
A distribution $ D$ on $ M$ is said to be locally constant if for every $ m\in M$ there is an open neighbourhood $ U$ of $ m$ such that $ dim (D_u)=k$ for all $ u\in U$ .
Is there any theorem which say that every involutive distribution is locally constant?
I think to use the Frobenius theorem which say that an involutive distribution is integrable, so the dimension of $ D$ is constant on each integral manifold, but the problem is that those integral manifold are not necessary open.
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