f-Eikonal helix submanifolds and f-Eikonal helix curves

Abstract

Let M{\subset}\mathbb{R}^{n} be a Riemannian helix submanifold with respect to the unit direction d{\in}\mathbb{R}^{n} and f:M{\to}\mathbb{R} be a eikonal function. We say that M is a f-eikonal helix submanifold if for each q{\in}M the angle between {\nabla}f and d is constant.Let M{\subset}\mathbb{R}^{n} be a Riemannian submanifold and {\alpha}:I{\to}M be a curve with unit tangent T. Let f:M{\to}\mathbb{R} be a eikonal function along the curve {\alpha}. We say that {\alpha} is a f-eikonal helix curve if the angle between {\nabla}f and T is constant along the curve {\alpha}. {\nabla}f will be called as the axis of the f-eikonal helix curve.The aim of this article is to give that the relations between f-eikonal helix submanifolds and f-eikonal helix curves, and to investigate f-eikonal helix curves on Riemannian manifolds.