Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2014; 51(2): 531-538

Printed March 31, 2014

https://doi.org/10.4134/BKMS.2014.51.2.531

Copyright © The Korean Mathematical Society.

A characterization of concentric hyperspheres in ${\mathbb R}^{n}$

Dong-Soo Kim and Young Ho Kim

Chonnam National University, Kyungpook National University

Abstract

Concentric hyperspheres in the $n$-dimensional Euclidean spa\-ce ${\mathbb R}^{n}$ are the level hypersurfaces of a radial function $f:{\mathbb R}^{n}\rightarrow {\mathbb R}$. The magnitude $|| \nabla f||$ of the gradient of such a radial function $f:{\mathbb R}^{n}\rightarrow {\mathbb R}$ is a function of the function $f$. We are interested in the converse problem. As a result, we show that if the magnitude of the gradient of a function $f:{\mathbb R}^{n}\rightarrow {\mathbb R}$ with isolated critical points is a function of $f$ itself, then $f$ is either a radial function or a function of a linear function. That is, the level hypersurfaces are either concentric hyperspheres or parallel hyperplanes. As a corollary, we see that if the magnitude of a conservative vector field with isolated singularities on $\mathbb R^n$ is a function of its scalar potential, then either it is a central vector field or it has constant direction.

Keywords: gradient, conservative vector field, central vector field, hypersurface, principal curvature, radial function

MSC numbers: 53A07