What is an example of a smooth submersion $P:S^{3}\to S^{2}$ for which the following statment is Not true:
For every vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ with the following properties:
- $P$ maps the solutions of $\tilde{X}$ to solutions of $X$
2.$Div(\tilde{X})$ is constant on each level set $P^{-1}(a),\;\;\forall a \in S^{2}$
Does the answer of this question depend on a particular metric on our manifolds $S^{2}$ and $S^{3}$, as divergence is a geometric quantity? I mean that if we change the metric, would the answer change?
The Note 1 of the following postfollowing post is a motivation for this question.
One can repeat the same question by replacing the above spheres with two compact reimannian manifolds $M,N$ of dimension $m>n$ respectively, such that every $m-n$ dimensional subbundle of $TM$ possess a nonvanishing smooth global section. Is such lifting $\tilde{X}$ always possible? Am I greedy if I ask for exsistence of a lifting $\tilde{X}$ with the above property with the additional condition $Div(\tilde{X})=Div(X)\circ P$. What type of obstruction would appear in this latest version?
To start, perhaps we need to have a zero- divergence non vanishing vector field tangent to $\ker DP$. Can we find such global vec. field?
