2011/10/10 by Eugenio Garnica, Garnica, Eugenio, Oscar Palmas +3 · 1 citation
Mathematics · #53B25 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53B25
paper · pdf · doi:10.48550/arxiv.1110.2201
Principal Direction, Constant Mean Curvature,Transnormal Functions, Closed conformal vector fields, 17 pages
arxiv created 2011/10/10 · arxiv updated 2011/10/12
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× N are either totally geodesic or cylinders.