On the problem of isometry of a hypersurface preserving mean curvature

Abstract

The problem of determining the {\it Bonnet hypersurfaces in} $R^{n+1}$, for $n>1$, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called {\it Bonnet associate} of the initial hypersurface. The orthogonal net which is called \hbox{\it $A$-net} is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in $R^{n+1}$ with no umbilical points is a Bonnet hypersurface if and only if it has an $A$-net.