  In this paper we discretize the incompressible Navier-Stokes equations in the
framework of finite element exterior calculus. We make use of the Lamb identity
to rewrite the equations into a vorticity-velocity-pressure form which fits
into the de Rham complex of minimal regularity. We propose a discretization on
a large class of finite elements, including arbitrary order polynomial spaces
readily available in many libraries. The main advantage of this discretization
is that the divergence of the fluid velocity is pointwise zero at the discrete
level. This exactness ensures pressure robustness. We focus the analysis on a
class of linearized equations for which we prove well-posedness and provide a
priori error estimates. The results are validated with numerical simulations.
