Higher codimensional Euclidean helix submanifolds

Abstract

A submanifold of Rⁿ whose tangent space makes constant angle with a fixed direction d is called a helix. Helix submanifolds are related with the eikonal PDE equation. We give a method to find every solution to the eikonal PDE on a Riemannian manifold locally. As a consequence we give a local construction of arbitrary Euclidean helix submanifolds of any dimension and codimension. Also we characterize the ruled helix submanifolds and in particular we describe those which are minimal.