Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2018; 55(6): 1823-1834

Online first article June 29, 2018      Printed November 30, 2018

https://doi.org/10.4134/BKMS.b171100

Copyright © The Korean Mathematical Society.

Euclidean submanifolds with conformal canonical vector field

Bang-Yen Chen, Sharief Deshmukh

USA, King Saud University

Abstract

The position vector field $\hbox{\bf x}$ 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 $\hbox{\bf x}^T$ of the position vector field is the most natural vector field tangent to the Euclidean submanifold $M$. We simply call the vector field $\hbox{\bf x}^T$ the \textit{canonical vector field} of the Euclidean submanifold $M$. In earlier articles \cite{C16,C17a,C17e,CV17,CW17}, we investigated Euclidean submanifolds whose canonical vector fields are concurrent, concircular, torse-forming, conservative or incompressible. 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.

Keywords: Euclidean submanifold, canonical vector field, conformal vector field, second fundamental form, umbilical, pseudo-umbilical

MSC numbers: 53A07, 53C40, 53C42