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.
Bang-Yen Chen, Sharief Deshmukh
USA, King Saud University
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
2017; 54(6): 2001-2011
2013; 50(4): 1061-1067
2002; 39(4): 671-680
2009; 46(6): 1141-1149
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd