I'll discuss existence problem for the the so-called 'cylinders' (to be defined in the talk) on projective manifolds. Featuring point will be a close relation between this problem and 2 celebrated ones, coming from arithmetic (integral points, heights, etc.) and differential geometry (Kobayashi hyperbolicity, Nevanlinna theory, etc.) This is in a sense a reminiscent of Vojta's approach to Mordell's conjecture. I'd like to apply this point of view on cylinders in the case of cubic hypersurfaces and highlight the proof that generic cubics (of any dimension) don't have cylinders at all. This is in line with a more general program, concerning the geometry of cubics, which I'll also outline (together with some questions).