Hypersurfaces with a canonical principal direction

Given a vector field X in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to X if the projection of X   onto the tangent space of the hypersurface gives a principal direction. We give different ways for building these hypersurfaces, as well as a number of useful characterizations. In particular, we relate them with transnormal functions and eikonal equations. With the further condition of having constant mean curvature (CMC) we obtain a characterization of the canonical principal direction surfaces in Euclidean space as Delaunay surfaces. We also prove that CMC constant angle hypersurfaces in a product R×NR×N are either totally geodesic or cylinders.