Necessary and sufficient condition for a surface to be a sphere

Author(s): Alexander G. Ramm1
1Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA.
Copyright © Alexander G. Ramm. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let \(S\) be a \(C^{1}\)-smooth closed connected surface in \(\mathbb{R}^3\), the boundary of the domain \(D\), \(N=N_s\) be the unit outer normal to \(S\) at the point \(s\), \(P\) be the normal section of \(D\). A normal section is the intersection of \(D\) and the plane containing \(N\). It is proved that if all the normal sections for a fixed \(N\) are discs, then \(S\) is a sphere. The converse statement is trivial.

Keywords: Conditions for a surface to be a sphere.

1. Introduction

Let \(S\) be a \(C^{1}\)-smooth closed connected surface in \(\mathbb{R}^3\), the boundaryof the domain \(D\), \(N=N_s\) be the unit outer normal to \(S\) at the point \(s\). Throughout we assume that \(S\) satisfies these assumptions. Let \(P\) be the normal section of \(D\). A normal section is the intersection of \(D\) and the plane containing \(N\). Our result is the following:

Theorem 1.1. If all the normal sections for a fixed \(N\) are discs, then \(S\) is a sphere. Conversely, if \(S\) is a sphere then all its normal sections are discs.

There are several “characterizations” of the sphere in the literature. We will use the following.

Lemma 1.2. Let \(r=r(p,q)\) be a parametric representation of \(S\). If \([r(p,q), N_s]=0\) for all \(s=s(p,q)\) on \(S\), then \(S\) is a sphere. Here \([r,N]\) is the vector product of two vectors.

A proof of this result can be found in [ 1, 2]. For convenience of the reader a short proof of Lemma 1.2 is given in Section 2.

2. Proof

Theorem 1.1. Let \(s\in S\) be a fixed point and \(P\) be one of the normal sections of \(D\) corresponding to \(N_s\). By assumption, this section is a disc. Let \(O\) be its center and \(R\) be its radius. Rotate \(P\) about \(N_s\). Each of the resulting normal sections is a disc of radius \(R\) centered at \(O\). If \(r=r(p,q)\) is a parametric representation of \(S\) then \([r, N]=0\) for every point of \(S\) because each such point belongs to a boundary of a disc centered at \(O\) with radius \(R\). From Lemma 1.2 it follows that \(S\) is a sphere.

Lemma 1.2. One has \(N=[r_p(p,q), r_q(p,q)]/|[r_p(p,q),r_q(p,q)]|\), where \([a,b]\) is the vector product of \(a\) and \(b\), and \(|a|\) is the length of the vector. Therefore \([r,N]=0\) implies \([r,[r_p(p,q),r_q(p,q)]]=0\) or \(r_p(r,r_q)- r_q(r,r_p)=0\), where \((a,b)\) is the scalar product of two vectors. The vectors \(r_p\) and \(r_q\) are linearly independent since the surface \(S\) is smooth. Thus, \((r,r_q)=0\) and \((r,r_p)=0\). Consequently \((r,r)=const\), that is, \(S\) is a sphere.

Competing Interests

The author declares that he has no competing interests.

References:

  1. Ramm, A. G. (2005). Inverse problems. Springer, New York.
  2. Ramm, A. G. (2013). The Pompeiu problem. Global Journ of Math. Anal., 1(1), 01-10. http://www.sciencepubco.com/index.php/GJMA/