Geodesic
A lemma on Lie bracket under insufficient smoothness
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 17 (2017) no. 1, pp. 73-77

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Let two vector fields on a $C^2$-variety $M$ be tangent to a $C^1$-submanifold $F\subset M$. We show if that these fields are differentiable at a point $p\in F$, then their Lie bracket is also tangent to $F$. This statement is a weakening of the “easy part” assumptions of the Frobenius theorem.
Keywords: Lie bracket. 
@article{VNGU_2017_17_1_a5,
     author = {K. V. Storozhuk},
     title = {A lemma on {Lie} bracket under insufficient smoothness},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {73--77},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2017_17_1_a5/}
}
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K. V. Storozhuk. A lemma on Lie bracket under insufficient smoothness. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 17 (2017) no. 1, pp. 73-77. http://geodesic.mathdoc.fr/item/VNGU_2017_17_1_a5/