I am working on path motion planning on different topological spaces. In order to prove the existence of some motion planning algorithms I would like to use that given an even dimensional sphere, we can always find a continuous vector field tangent to it which only vanishes at one point.
My attempts:
First of all I think that that statement should be true since we could use the stereographic projection (which is a diffeomorphism) in order to "bring" or "take back" a smooth vector field defined on $\mathbb{R}^{2n}$ (assuming we are dealing with $\mathbb{S}^{2n}$) which only vanishes at one point.
But I am doing a project about other topic and if I use that, then I should introduce and define charts, diffeomorphisms...etc and I am restricted on the number of pages. That is the reason why I am looking for other argument, moreover, I don't need smoothness so that would be like killing a mosquito with a cannon ball.
What about using that the stereographic projection is an homeomorphism and using the same idea as before? Continuity is preserved by homeomorphisms so I would have finished. But again if could be possible I would like to avoid using stereographic projection.
What I have consulted:
My question:
What other idea do you suggest? Is there an intuitive vector field that works and I am missing it?
Thanks in advance!
It shouldn't be too hard to construct such a vector field. The hard trick is proving that all vectors fields on $S^{2n}$ vanish at some point.
Start with the tangent vectors to a rotation about some axis. This provides a vector field with exactly two $0$ points. Now morph the sphere below to bring these two points together. Everwhere off the joined point, the vector field has undergone a continuous transformation, so it remains a vector field. At the joined point, the behavior is singular, but since the vector field vanishes there, the singularity is smoothed out.