Podousova T. LGT-grid and its deformations

It is proved the existence of actual grid of lines of the geodesic torsion (LGT-grid) on any regular surface and are studied its properties. We study A-deformations of a regular surface with stationary lengths of LGT-lines , find the conditions of existence of such deformations; investigate A-deformations of regular surface with -a stationary LGT-grid, - a stationary complete geodesic torsion, - a stationary invariant of grid tensor, and find the conditions of existence for such deformations. For each of these tasks, we formulate theorems whose contain the conditions of existence and arbitrariness of their solution the tensor fields are presented in these conditions in an explicit form. Theoretical results are illustrated on examples of surfaces.