## 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 $$E^3$$.

### MSC:

 52B70 Polyhedral manifolds 57R42 Immersions in differential topology