TITLE: A non integrable distribution which is totally geodesic
QUESTION [7 upvotes]: Is there a non integrable  $2$ dimensional  distribution $D$ of a $3$ dimensional  Riemannian manifold such that the distribution is totally geodesic in the following sense:
Every geodesic whose tangent vector of its intitial  point is tangent to the distribution then the tangent vector  at all its points is tangent to $D$, too.

REPLY [2 votes]: Take $\mathbb{R}^3$ with the distribution which is the kernel of the one-form $dz - y dx$. This is the standard example of a contact structure on $\mathbb{R}^3$. See https://en.wikipedia.org/wiki/Contact_geometry .