Piecewise distance preserving maps

Abstract

These lectures were a part of the geometry course held during the Fall 2011 Mathematics Advanced Study Semesters (MASS) Program at Penn State (http://www.math.psu.edu/mass/).The lectures are meant to be accessible to advanced undergraduate and early graduate students in mathematics. We have placed a great emphasis on clarity and exposition, and we have included many exercises. Hints and solutions for most of the exercises are provided in the end.The lectures discuss piecewise distance preserving maps from a 2-dimensional polyhedral space into the plane. Roughly speaking, a polyhedral space is a space that is glued together out of triangles, for example the surface of a polyhedron. If one imagines such a polyhedral space as a paper model, then a piecewise distance preserving map into the plane is essentially a way to fold the model so that it lays flat on a table. We have five lectures on the following topics:⋄ Zalgaller's folding theorem, which guarantees the existence of a piecewise distance preserving map from a 2-dimensional polyhedral space into the plane. In other words, it is always possible to fold the paper model onto the table. ⋄ Brehm's extension theorem, which allows one to build piecewise distance preserving maps from a convex polygon into the plane with prescribed images on a finite subset of the polygon. ⋄ Akopyan's approximation theorem, which allows one to approximate maps from a 2-dimensional polyhedral space into the plane by piecewise distance preserving maps. ⋄ Gromov's rumpling theorem, which shows the existence of a length-preserving map from the sphere into the plane; i.e., a map that preserves the lengths of all curves. ⋄ An entertaining problem of Arnold on paper folding, which asks if it is possible to fold a square in the plane so that the perimeter increases.3 4We only consider the 2-dimensional case to keep things easy to visualize. However, most of the results admit generalizations to higher dimensions. These results are discussed in the Final Remarks, where proper credit and references are given.Acknowledgments. We would like to thank Arseniy Akopyan, Robert Lang, Alexei Tarasov for their help. Also we would like to thank all the students in our class for their participation and true interest.

Local characterization of polyhedral spaces

“…In full generality this result was proved recently by Akopyan [1], his proof is based on earlier results obtained by Zalgaller [8] and Krat [3]. Akopyan's proof is sketched in the lecture notes of the second author [7].…”

A constructive version of the extremum value theorem for spaces of vector-valued functions

“…In contrast, f (x) = x + π is not an algebraic motion. Any three algebraic points forming a non-degenerate triangle can be moved by an algebraic motion to new algebraic points preserving the respective distances (Petrunin and Yashinski [8]). An algebraic piecewise motion f on a bounded algebraic two-dimensional polyhedron P is a pair of a triangulation {T i } i of the polyhedron and a collection of algebraic motions {f i } i such that f | Ti = f i .…”

Approximating continuous maps by isometries

“…The proof goes through nearly unchanged, and can be found in [Ako07] (in Russian). An English proof can be found in [Min13], and the case when n = 2 can be found in [PY15]. Let (X , T ) be a polyhedron (that is, a topological space X with a locally finite triangulation T ) and let x ∈ X .…”