f-Eikonal helix submanifolds and f-Eikonal helix curves

TL;DR

This paper explores the properties and relationships of f-eikonal helix submanifolds and curves in Riemannian manifolds, focusing on the constancy of angles between the gradient of an eikonal function and specific directions or tangent vectors.

Contribution

It introduces the concepts of f-eikonal helix submanifolds and curves, establishing their relationships and analyzing their properties in Riemannian geometry.

Findings

Characterization of f-eikonal helix submanifolds

Conditions for curves to be f-eikonal helix curves

Relations between the gradient of f and geometric structures

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.