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For a manifold $M$, a smooth subbundle of the tangent bundle $TM$ is called a distribution on $M$. A distribution $D$ is said to be \textsl{bracket generating} if successive Lie brackets of (smooth) local vector fields in $D$
generate the whole tangent bundle. By Chow's theorem, any two points of M
can be joined by a smooth path which is tangential to the distribution at all points. We shall begin by explaining this theorem and some related questions in the context of 3-dimensional contact distributions, which are the primary examples of bracket generating distributions.
More generally, one may consider smooth immersions or embeddings $f:\Sigma\to (M,D)$ of an arbitrary manifold $\Sigma$ such that the derivative of $f$ maps $T\Sigma$ into $D$. Such maps are called horizontal to $D$. Horizontal immersions to contact distributions are completely understood due to a result of Gromov. We also have a similar classification of smooth horizontal immersions for holomorphic contact structures and their real analogues. This is a joint work with Aritra Bhowmick. |