The speaker aims to propose several unsolved problems concerning the irrotational oscillation of a water droplet under zero gravity. The governing equation of this physical system is derived and converted to a quasilinear dispersive partial differential equation defined on the sphere, which formally resembles the capillary water waves equation but is defined on a curved manifold instead. A coordinate-independent, global para-differential calculus defined on compact Lie groups and homogeneous spaces will be developed as a toolbox regarding this equation. Three types of mathematical problems related to this model will be discussed, in observation of hydrodynamical experiments under zero gravity: (1) Local Existence and Strichartz type inequalities for the linearized problem (2) existence of periodic solutons (3) normal form reduction and generic lifespan estimate. It is pointed out that all of these problems are closely related to certain Diophantine equations, especially the third one.