关于保持距离映射的Aleksandrov问题

On Conservative Mapping of Aleksandrov Problem

中文关键词:  保$\rho$ 映射  强保$\rho$ 映射  等距映射  Aleksandrov问题.

英文关键词:distance-$\rho$ preserving mapping  strongly distance-$\rho$ preserving mapping  isometry mapping  Aleksandrov problem.

中文摘要:

      设$E,F$为实Hilbert空间,$f:E\rightarrow F$为一映射.本文证明了在一定条件下,若$f$保距离$a$和$b$ (其中, $a,b$为两个正实数),则$f$为一个仿射等距.从而部分地解决了Aleksandrov在1970年提出的Aleksandrov问题.

英文摘要:

      Let $E$ and $F$ be real Hilbert spaces, and $f:E\rightarrow F$ be a mapping. This paper shows that $f$ is an affine isometry if $f$ preserves distance $a$ and $b$ (where $a, b$ are two positive real numbers) and satisfies some additional conditions, and hence partly answers the Aleksandrov problem.