• Received : 2012.12.24
  • Published : 2014.03.31


Concentric hyperspheres in the n-dimensional Euclidean space $\mathbb{R}^n$ 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.


gradient;conservative vector field;central vector field;hypersurface;principal curvature;radial function


  1. R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II, Partial differential equations, Reprint of the 1962 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1989.
  2. L. A. Caffarelli and M. G. Crandall, Distance functions and almost global solutions of eikonal equations, Comm. Partial Differential Equations 35 (2010), no. 3, 391-414.
  3. M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ, 1976.
  4. T. L. Chow, Mathematical Methods for Physicists: A concise introduction, Cambridge University Press, Cambridge, 2000.
  5. M.W. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra, Pure and Applied Mathematics, Vol. 60, Academic Press, New York-London, 1974.
  6. D. Khavinson, A note on entire solutions of the eiconal [eikonal] equation, Amer. Math. Monthly 102 (1995), no. 2, 159-161.
  7. K. Nomizu, Elie Cartan's work on isoparametric families of hypersurfaces, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973), pp. 191-200, Amer. Math. Soc., Providence, R.I., 1975.
  8. O. N. Stavroudis and R. C. Fronczek, Caustic surfaces and the structure of the geometrical image, J. Opt. Soc. Amer. 66 (1976), no. 8, 795-800.
  9. J. A. Thorpe, Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1979.

Cited by

  1. Various centroids and some characterizations of catenary rotation hypersurfaces vol.42, pp.13036149, 2018,