American Institute of Mathematical Sciences

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  2016, 23: 1-7. doi: 10.3934/era.2016.23.001

Extensions of isometric embeddings of pseudo-Euclidean metric polyhedra

Pavel Galashin 1, and  Vladimir Zolotov 2,

1. 

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, United States
2. 

Mathematics and Mechanics Faculty, St. Petersburg State University, Universitetsky pr., 28, Stary Peterhof, 198504, Russian Federation

Received  October 2015 Published  January 2016

We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible dimension. We provide a simple algorithm for constructing such embeddings. We also show that every partial simplicial isometric embedding of such space in general position extends to a simplicial isometric embedding of the whole space.
Keywords: metric geometry, pseudo-metric polyhedra, isometric extension, isometric simplicial embedding, Minkowski space., Discrete geometry.
Mathematics Subject Classification: 51F9.
Citation: Pavel Galashin, Vladimir Zolotov. Extensions of isometric embeddings of pseudo-Euclidean metric polyhedra. Electronic Research Announcements, 2016, 23: 1-7. doi: 10.3934/era.2016.23.001
References:
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V. A. Zalgaller, Isometric imbedding of polyhedra, Dokl. Akad. Nauk SSSR, 123 (1958), 599-601.  Google Scholar

show all references

References:
[1]

A. V. Akopyan and A. Tarasov, PL-analogue of the Nash-Kuiper Theorem, preprint, 2007. Google Scholar

[2]

U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on $\mathbbR^m$, J. Geom., 16 (1981), 187-193. doi: 10.1007/BF01917587.  Google Scholar

[3]

Yu. D. Burago and V. A. Zalgaller, Isometric piecewise-linear embeddings of two-dimensional manifolds with a polyhedral metric into $\mathbbR^3$, Algebra i Analiz, 7 (1995), 76-95.  Google Scholar

[4]

G. H. Golub and C. F. Van Loan, Matrix Computations, Fourth edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013.  Google Scholar

[5]

S. A. Krat, Approximation Problems in Length Geometry, Ph.D. Thesis, The Pennsylvania State University, ProQuest LLC, Ann Arbor, MI, 2004.  Google Scholar

[6]

B. Minemyer, Simplicial Isometric Embeddings of Indefinite Metric Polyhedra, arXiv:1211.0584, 2015. Google Scholar

[7]

V. A. Zalgaller, Isometric imbedding of polyhedra, Dokl. Akad. Nauk SSSR, 123 (1958), 599-601.  Google Scholar

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