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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.
Yes, the standard contact structure on the unit three-sphere in $\mathbb{R}^4 = \mathbb{C}^2$, for instance. The Legendrian great circles are the intersections of the sphere with the Lagrangian two-planes.
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 .
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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.
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Yes, the standard contact structure on the unit three-sphere in $\mathbb{R}^4 = \mathbb{C}^2$, for instance. The Legendrian great circles are the intersections of the sphere with the Lagrangian two-planes.
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