  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
a congruence of arbitrary timelike curves on a given manifold. In this approach
Einstein's equations do not play any role and this can be regarded as an {\it
off-shell} description of fluid-gravity correspondence. We argue that {\it our
metric effectively encapsulates the information of forcing terms in the
governing equations as if a free fluid is parallel transported on this curved
background}. We finally discuss the implications of this interesting
observation and its potentiality in helping us to understand hydrodynamical
flows in a probable new setting.
