Pogorelov's problem on isometric transformations of a cylindrical surface

In the late 1960s A. V. Pogorelov considered the problem of piecewise smooth isometric embeddings of the surface of a right circular cylinder in three-dimensional Euclidean space with the so-called condition of support on the edges: it is assumed that the circular components of the cylinder boundary are embedded in a standard way – as circles located in parallel planes one above the other. This problem is motivated by the applied problem from the theory of shells on the deformation of a thin-walled cylindrical pipe subjected a strong (supercritical) compression along the axis. A. V. Pogorelov in his book “Geometric Methods in the Nonlinear Theory of Elastic Shells” of 1967 claims that he solved the problem of the existence of a non-trivial isometric embedding of a cylindrical surface under these conditions, and presented several possible options for such an embedding. This result was used by A. V. Pogorelov in the analysis of the mechanical properties of the supercritical elastic state of a cylindrical shell.

In work [1] by means of some results of previous works [2] and [3], M. I. Shtogrin showed that Pogorelov's reasoning contains gaps: 1967 book, chapter 8, paragraph 4. In work [1], it is proved in detail that embedded surfaces presented by A. V. Pogorelov are not isometric to the cylinder. Developing this research, M. I. Shtogrin has constructed non-trivial isometric embeddings of the cylinder, which satisfy Pogorelov's conditions, in the class of piecewise smooth surfaces with an unbounded set of smooth pieces. The first example of such an embedding can be obtained if the surface depicted in a Fig. 15c in [3] is first cut into two congruent parts along a circle (parallel to the bases) and then a new surface is glue of these parts with the cut circles on the border. However, the existence of a non-trivial isometric embedding of the cylinder with finite number of smooth pieces is not established yet and in this case the discussed Pogorelov's problem remains unsolved.

References

1. M. I. Shtogrin, “Zadacha Pogorelova ob izometricheskikh preobrazovaniyakh tsilindricheskoi poverkhnosti”, UMN, 74:6(450) (2019), 169–170     [M. I. Shtogrin, “Pogogrelov's problem on isometric transformations of a cylindrical surface”, Uspekhi Mat. Nauk, 74:6(450) (2019), 169–170    ]
2. M. I. Shtogrin, “Ob odnoi zadache Pogorelova”, UMN, 73:1(439) (2018), 185–186          ; Russian Math. Surveys, 73:1 (2018), 178–180      
3. M. I. Shtogrin, “Izometricheskie pogruzheniya konusa i tsilindra”, Izv. RAN. Ser. matem., 73:1 (2009), 187–224            ; Izv. Math., 73:1 (2009), 181–213