Let $V$ be a vector field on $S^3$ such that its singularity points, namely the points at which the vector field vanishes, are only sinks or sources (i.e. the field is converging or diverging). Is there an upper bound on the number $N$ of sinks and sources? Is this number 2? Is it possible that N=1?
There are examples of non-singular fields (Hopf fibering) for which N=0 and simple examples with N=2 (gradient of height function). I am asking if there are other possibilities.
The analogous question on $S^2$ has negative answer by the Poincaré-Hopf theorem which states that N=2.
Here is my attempt: If there is an integral curve of the vector field which joins a source A with a sink B then every integral curve from A must end in B (intuitively, this follows from an argument on connectedness of the boundary of a small ball centered at A) thus in this case N=2, and we are left with the possibility that all the sinks and sources are 'separated' by invariant sets, but is this really possible?