$A$-deformations of a surface with stationary lengths of $LGT$-lines

Abstract

We investigate infinitesimal areal deformations ($A$-deformations) of the first order under which the lengths of $LGT$-lines of a surface are preserved in the $E_3$ -space. We prove that any regular surface of the class $C^4$ of nonzero Gaussian curvature without umbilical points admits nontrivial $A$-deformations with stationary lengths of LGT-lines.