Motivated by the Navier-Stokes equations, which are a set of unsolved equations related to fluid motion in R^3, we explored the incompressibility condition and the Neumann boundary problem. After exploring, we noticed that using iterated Riesz transforms of the boundary data could be used to get information about the velocity field directly from the boundary data. We then used this same logic to look at the same problem in a spherical space, and found no definition for the Riesz transforms. By using the incompressibility condition and solving the Neumann boundary problem on a sphere, using both separation of variables and potential theory, we can define the Riesz transforms of a function on the sphere. This allows us to use boundary data on a sphere to describe the divergence-free field, or in the case of the incompressible Navier-Stokes equation, allows us to solve for the velocity field given initial data.