Euclidean submanifolds with conformal canonical vector field

TL;DR

This paper characterizes Euclidean submanifolds whose canonical vector field is conformal, exploring their properties, applications, and providing global results for complete cases, thus advancing understanding of geometric structures in Euclidean spaces.

Contribution

It introduces a characterization of Euclidean submanifolds with conformal canonical vector fields and presents new global results for complete submanifolds.

Findings

Characterization of Euclidean submanifolds with conformal canonical vector fields

Applications of the conformal property in geometric analysis

Global results for complete Euclidean submanifolds

Abstract

The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold $M$. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the Euclidean submanifold $M$. We simply call the vector field x^T the \textit{canonical vector field} of the Euclidean submanifold M. In earlier articles, we investigated Euclidean submanifolds whose canonical vector fields are concurrent, concircular, or torse-forming. In this article we study Euclidean submanifolds with conformal canonical vector field. In particular, we characterize such submanifolds. Several applications are also given. In the last section we present three global results on complete Euclidean submanifolds with conformal canonical vector field.