Non-Metric Affine Geometry and Modified Navier–Stokes Equations

We develop a geometric formulation of incompressible hydrodynamics on manifolds equipped with a torsion-free Weyl affine connection with pure-trace non-metricity. Starting from a covariant variational principle, we derive modified Navier–Stokes equations that include explicit curvature and non-metric correction terms. We compute the full affine Ricci tensor in dimension three, establish a Weyl-adapted divergence theorem, and obtain a closed-form expression for the viscous dissipation operator. The resulting system preserves incompressibility, admits an energy inequality, and reduces to the classical Navier–Stokes equations in the metric limit. We further prove local well-posedness in Sobolev spaces and discuss connections to geometric hydrodynamics, turbulence modeling, and modified gravity frameworks.