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Hypersurfaces with a canonical principal direction. (English) Zbl 1251.53012
Summary: 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 additional 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.

53B25 Local submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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