On Invariants of Surfaces with Isometric on Sections
Annotatsiya
In one of the directions of classical differential geometry, the properties of geometric objects are studied in their entire range, which is called geometry "in large". Many problems of geometry "in large" are connected with the existence and uniqueness of surfaces with given characteristics. Geometric features can be intrinsic curvature, extrinsic or Gaussian curvature, and other features associated with the surface. The existence of a polyhedron with given curvatures of vertices or with a given development is also a problem of geometry "in large". Therefore, the problem of finding invariants of polyhedra of a certain class and the solution of the problem of the existence and uniqueness of a polyhedra with given values of the invariant are relevant. This work is devoted to finding invariants, surfaces isometric on sections. In particular, we study the expansion properties of convex polyhedra that preserve isometry on sections. For such polyhedra, an invariant associated with the vertex of a convex polyhedral angle is found. Using this invariant, we can consider the question of restoring a convex polyhedron with given values of conditional curvature at the vertices. The isometry on section differs from the isometry of surfaces. The isometry of surfaces does not imply the isometry in sections, and vice versa. One of the invariants of surfaces isometric in cross sections is the area of the cylindrical image. This paper presents the properties of the area of a cylindrical image.