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Let $(M.g)$ be a Riemannian manifold and $D$ is a distribution on $M$. Assume that $D$ is invariant under the action of the isometry group of $M$.

Under which conditions such $D$ is an integrable distribution?What is an example of a non integrable distribution with this invariant property?

In the case of integrability, under what conditions the leaves of corresponding foliation are totally geodesic?

The motivation comes from the case $\mathbb{R}^{n} \setminus \{0\}$ with the standard metrics. there is only one distribution with this property, which is integrable.

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  • $\begingroup$ Take any non-integrable $D$, and perturb any $g$ to have trivial isometry group... $\endgroup$
    – Francois Ziegler
    Mar 8, 2017 at 8:23
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    $\begingroup$ The plane is maybe not so relevant here, since all rank 1 distributions are integrable. $\endgroup$
    – Ben McKay
    Mar 8, 2017 at 8:34
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    $\begingroup$ You might look at the Berger metrics on the odd dimensional spheres: en.wikipedia.org/wiki/Berger%27s_sphere $\endgroup$
    – Ben McKay
    Mar 8, 2017 at 8:36
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    $\begingroup$ @AliTaghavi $\rho(x,y)(dx^2+dy^2)$ where $\rho>0$ has no nontrivial symmetries? Metrics without isometries are generic. $\endgroup$
    – Francois Ziegler
    Mar 8, 2017 at 8:48
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    $\begingroup$ Your questions remind me the topic of "polar actions". A $G$-action (by isometries) is called polar (or better) locally polar if the distribution (on the open set of principal orbits) of normal spaces is integrable. Most $G$-actions are not polar. For example most representations of a compact Lie group induce a non polar action on a sphere. Dadok's theorem tells that the polar ones are those having the same orbits of a so called s-representation. For this topic see Bernd-Console-Olmos' book "Submanifolds and holonomy". $\endgroup$
    – Holonomia
    Mar 11, 2017 at 10:12

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