Isometric piecewise-linear immersions of two-dimensional manifolds with polyhedral metric into \(\mathbb{R}^ 3\). (English. Russian original) Zbl 0851.52018

St. Petersbg. Math. J. 7, No. 3, 369-385 (1996); translation from Algebra Anal. 7, No. 3, 76-95 (1995).
The main result of this interesting paper is an isometric immersion (or embedding) theorem for polynomial 2-manifolds in \(E^3\). More precisely, if one starts with a polyhedral metric on a surface \(M\) and any \(C^2\)-smooth immersion (or embedding) into \(E^3\) which is contracting with respect to this metric, then it is possible to find a \(C^0\)-approximation by isometric piecewise linear immersions (or embeddings). As a surprising example, one obtains flat polyhedral tori approximating the standard torus in \(E^3\). As a surprising corollary, one obtains that every polyhedral metric on a surface admits an isometric piecewise linear immersion (or embedding) into \(E^3\).


52B70 Polyhedral manifolds
57R42 Immersions in differential topology