Extensions of isometric embeddings of pseudo-Euclidean metric polyhedra

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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

Abstract
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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:

[1]	A. V. Akopyan and A. Tarasov, PL-analogue of the Nash-Kuiper Theorem, preprint, 2007.
[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.
[3]	Yu. D. Burago and V. A. Zalgaller, Isometric piecewise-linear embeddings of two-dimensional manifolds with a polyhedral metric into $\mathbb{R}^3$, Algebra i Analiz, 7 (1995), 76-95.
[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.
[5]	S. A. Krat, Approximation Problems in Length Geometry, Ph.D. Thesis, The Pennsylvania State University, ProQuest LLC, Ann Arbor, MI, 2004.
[6]	B. Minemyer, Simplicial Isometric Embeddings of Indefinite Metric Polyhedra, arXiv:1211.0584, 2015.
[7]	V. A. Zalgaller, Isometric imbedding of polyhedra, Dokl. Akad. Nauk SSSR, 123 (1958), 599-601.

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References:

[1]	A. V. Akopyan and A. Tarasov, PL-analogue of the Nash-Kuiper Theorem, preprint, 2007.
[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.
[3]	Yu. D. Burago and V. A. Zalgaller, Isometric piecewise-linear embeddings of two-dimensional manifolds with a polyhedral metric into $\mathbb{R}^3$, Algebra i Analiz, 7 (1995), 76-95.
[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.
[5]	S. A. Krat, Approximation Problems in Length Geometry, Ph.D. Thesis, The Pennsylvania State University, ProQuest LLC, Ann Arbor, MI, 2004.
[6]	B. Minemyer, Simplicial Isometric Embeddings of Indefinite Metric Polyhedra, arXiv:1211.0584, 2015.
[7]	V. A. Zalgaller, Isometric imbedding of polyhedra, Dokl. Akad. Nauk SSSR, 123 (1958), 599-601.

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