Inspired by differential equation $$y(1+y'^2)=c$$
which generates the cycloid we consider the following differential equation on a Riemannian manifold:
$$f(1+|\nabla f|^2)=c$$
On the other hand the classical cycloid has some singularities(non smooth points).
This motivates the following question:
Is there an example of a Riemannian manifold which admits a globally smooth non constant cycloid function?
