Piecewise distance preserving maps

Petrunin, A.; Yashinski, A.

doi:10.1090/spmj/1381

Piecewise distance preserving maps
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by A. Petrunin and A. Yashinski
Translated by: the authors

St. Petersburg Math. J. 27 (2016), 155-175

DOI: https://doi.org/10.1090/spmj/1381

Published electronically: December 7, 2015

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Abstract:

This is part of a geometry course held during the Fall 2011 MASS Program at Penn State (www.math.psu.edu/mass/). The online version of these lectures also contains video illustrations, hints, and solutions for most of the exercises, and a minimalistic section covering preliminaries.

In the lectures, piecewise distance preserving maps are discussed that act 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.

Only the 2-dimensional case is considered, 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.

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