The incompressible Navier-Stokes (NS) equation is known to govern the hydrodynamic limit of essentially any fluid and its rich non-linear structure has critical implications in both mathematics and physics. The employability of the methods of Riemannian geometry to the study of hydrodynamical flows has been previously explored from a purely mathematical perspective. In this work, we propose a bulk metric in $(p+2)$-dimensions with the construction being such that the induced metric is flat on a timelike $r = r_c$ (constant) slice. We then show that the equations of {\it parallel transport} for an appropriately defined bulk velocity vector field along its own direction on this manifold when projected onto the flat timelike hypersurface requires the satisfaction of the incompressible NS equation in $(p+1)$-dimensions. Additionally, the incompressibility condition of the fluid arises from a vanishing expansion parameter $\theta$, which is known to govern the convergence (or divergence) of 查看全文>>