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Integrability of distributions which are invariant under isometry group

0

$begingroup$

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 $mathbbR^n setminus 0$ with the standard metrics. there is only one distribution with this property, which is integrable.

differential-geometry foliations

share|cite|improve this question

edited Mar 23 at 12:46

Ali Taghavi

asked Mar 23 at 3:20

Ali TaghaviAli Taghavi

263430

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Do you mean to restrict, e.g., to metrics for which the isometry action on $M$ is transitive? Without some restriction, one can construct trivial nonintegrable examples by choosing $g$ with trivial isometry group and choosing any nonintegrable distribution on $M$.
  $endgroup$
  – Travis
  Mar 23 at 18:22
  

  
  
  
  

add a comment | 

0

$begingroup$

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 $mathbbR^n setminus 0$ with the standard metrics. there is only one distribution with this property, which is integrable.

differential-geometry foliations

share|cite|improve this question

edited Mar 23 at 12:46

Ali Taghavi

asked Mar 23 at 3:20

Ali TaghaviAli Taghavi

263430

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Do you mean to restrict, e.g., to metrics for which the isometry action on $M$ is transitive? Without some restriction, one can construct trivial nonintegrable examples by choosing $g$ with trivial isometry group and choosing any nonintegrable distribution on $M$.
  $endgroup$
  – Travis
  Mar 23 at 18:22
  

  
  
  
  

add a comment | 

$begingroup$

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 $mathbbR^n setminus 0$ with the standard metrics. there is only one distribution with this property, which is integrable.

differential-geometry foliations

share|cite|improve this question

edited Mar 23 at 12:46

Ali Taghavi

asked Mar 23 at 3:20

Ali TaghaviAli Taghavi

263430

$endgroup$

share|cite|improve this question

edited Mar 23 at 12:46

Ali Taghavi

asked Mar 23 at 3:20

Ali TaghaviAli Taghavi

263430

share|cite|improve this question

edited Mar 23 at 12:46

Ali Taghavi

asked Mar 23 at 3:20

Ali TaghaviAli Taghavi

263430

asked Mar 23 at 3:20

Ali TaghaviAli Taghavi

263430

asked Mar 23 at 3:20

Ali TaghaviAli Taghavi

263430