Hypersurfaces with a canonical principal direction

TL;DR

This paper studies hypersurfaces with a canonical principal direction relative to a vector field, providing construction methods, characterizations, and specific classifications in Euclidean space and product manifolds.

Contribution

It introduces new characterizations and construction methods for hypersurfaces with a canonical principal direction, linking them with transnormal functions and eikonal equations.

Findings

Canonical principal direction hypersurfaces relate to transnormal functions.

CMC hypersurfaces with a canonical principal direction are Delaunay surfaces in Euclidean space.

CMC constant angle hypersurfaces in R×N are either totally geodesic or cylindrical.

Abstract

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 $\mathbb{R}\times N$ are either totally geodesic or cylinders.