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        A non integrable distribution which is totally geodesic

        up vote 14 down vote favorite

        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.

        2 Answers

        up vote 4 down vote accept

        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.

        up vote 3 down vote

        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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        A non integrable distribution which is totally geodesic

        up vote 14 down vote

        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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        up vote 9 down vote

        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.

        edit

        @AliTaghavi. it is a good question. Some years ago Patrick Massot, a student of Giroux, did some nice work on this: projecteuclid.org/euclid.gt/1513800108 - alvarezpaiva Jul 26 at 14:37

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