부산대학교 도서관

Let $B^n(r)$ be the closed unit ball of radius $r>0$ in ${\bf R}^n$. It is shown that if $B^n(1)$ is immersed smoothly and isometrically in ${\bf R}^m$ with $m<2n$, then the diameter of the image of $B^n(1)$ is at least 1. The general case is nicely motivated by the case $n=2$, $m=3$, but the general proof requires a nontrivial extension of the techniques involved in the special case. If $m\geq 2n$, then there is a smooth isometric immersion of $B^n(1)$ into $B^m(r)$ for any $r>0$ (confinability) [see E. M. Kramer and T. A. Witten, Phys. Rev. Lett. {\bf 78} (1997), no. 7, 1303--1306].