Euclidean submanifolds with conformal canonical vector field
Bull. Korean Math. Soc. 2018 Vol. 55, No. 6, 1823-1834 https://doi.org/10.4134/BKMS.b171100 Published online June 29, 2018 Printed November 30, 2018
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
