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arXiv · 1110.2201

Hypersurfaces with a canonical principal direction

Eugenio Garnica·Oscar Palmas·Gabriel Ruiz-Hernández
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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.

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Eugenio Garnica, Oscar Palmas, Gabriel Ruiz-Hernández. 2011-10-10. Hypersurfaces with a canonical principal direction. https://arxiv.org/abs/1110.2201

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