We have developed dynamic manifold solutions for the Navier-Stokes equations using an extension of differential geometry called the calculus for moving surfaces. Specifically, we have shown that the geometric solutions to the Navier-Stokes equations can take the form of fluctuating spheres, constant mean curvature surfaces, generic wave equations for compressible systems, and arbitrarily curved shapes for incompressible systems in various scenarios. These solutions apply to predominantly incompressible and compressible systems for the equations in any dimension, while the remaining cases are yet to be solved. We have demonstrated that for incompressible Navier-Stokes equations, geometric solutions are always bound by the curvature tensor of the closed smooth manifold for every smooth velocity field. As a result, solutions always converge for systems with constant volumes.

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