Classification of constant angle hypersurfaces in warped products via eikonal functions

TL;DR

This paper classifies constant angle hypersurfaces in warped products, linking their geometry to solutions of a generalized eikonal equation and showing minimal ones are cylindrical over minimal submanifolds.

Contribution

It extends classification results of constant angle hypersurfaces to warped products and introduces a geometric method to solve the associated eikonal equation.

Findings

Constant angle hypersurfaces are classified in warped products.

Solutions to the generalized eikonal equation are obtained geometrically.

Minimal constant angle hypersurfaces are cylinders over minimal submanifolds.

Abstract

Given a warped product of the real line with a Riemannian manifold of arbitrary dimension, we classify the hypersurfaces whose tangent spaces make a constant angle with the vector field tangent to the real direction. We show that this is a natural setting in which to extend previous results in this direction made by several authors. Moreover, when the constant angle hypersurface is a graph over the Riemannian manifold, we show that the function involved satisfies a generalized eikonal equation, which we solve via a geometric method. In the final part of this paper we prove that minimal constant angle hypersurfaces are cylinders over minimal submanifolds.