Geodesic
Isometric immersions of a cone and a cylinder
Izvestiya. Mathematics , Tome 73 (2009) no. 1, pp. 181-213

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We thoroughly analyse the method used by Pogorelov to construct piecewise-smooth tubular surfaces in $\mathbb R^3$ isometric to the surface of a right circular cylinder. The properties of the inverse images of edges of any tubular surface on its planar unfolding are investigated in detail. We find conditions on plane curves lying on the unfolding that enable them to be the inverse images of edges of some tubular surface. We make a refinement concerning the number of smooth pieces that form a piecewise-smooth tubular surface. We generalize Pogorelov's method from the surface of a right circular cylinder to that of a right circular cone.
Keywords: surface theory, surfaces in three-dimensional space. 
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     author = {M. I. Shtogrin},
     title = {Isometric immersions of a cone and a cylinder},
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     language = {en},
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M. I. Shtogrin. Isometric immersions of a cone and a cylinder. Izvestiya. Mathematics , Tome 73 (2009) no. 1, pp. 181-213. http://geodesic.mathdoc.fr/item/IM2_2009_73_1_a9/