  We investigate a projection-based reduced-order model of the steady
incompressible Navier-Stokes equations for moderate Reynolds numbers. In
particular, we construct an "embedded" reduced basis space, by applying proper
orthogonal decomposition to the Shifted Boundary Method, a high-fidelity
embedded method recently developed. We focus on the geometrical parametrization
through level-set geometries, using a fixed Cartesian background geometry and
the associated mesh. This approach avoids both remeshing and the development of
a reference domain formulation, as typically done in fitted mesh finite element
formulations. Two-dimensional computational examples for one and
three-parameter dimensions are presented to validate the convergence and the
efficacy of the proposed approach.
