Isometric Embeddings of Pro-Euclidean Spaces

Barry Minemyer

Analysis and Geometry in Metric Spaces (2015)

  • Volume: 3, Issue: 1, page 317-324, electronic only
  • ISSN: 2299-3274

Abstract

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In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].

How to cite

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Barry Minemyer. "Isometric Embeddings of Pro-Euclidean Spaces." Analysis and Geometry in Metric Spaces 3.1 (2015): 317-324, electronic only. <http://eudml.org/doc/276005>.

@article{BarryMinemyer2015,
abstract = {In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].},
author = {Barry Minemyer},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {differential geometry; discrete geometry; metric geometry; Euclidean polyhedra; polyhedral space; intrinsic isometry; polyhedral space},
language = {eng},
number = {1},
pages = {317-324, electronic only},
title = {Isometric Embeddings of Pro-Euclidean Spaces},
url = {http://eudml.org/doc/276005},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Barry Minemyer
TI - Isometric Embeddings of Pro-Euclidean Spaces
JO - Analysis and Geometry in Metric Spaces
PY - 2015
VL - 3
IS - 1
SP - 317
EP - 324, electronic only
AB - In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].
LA - eng
KW - differential geometry; discrete geometry; metric geometry; Euclidean polyhedra; polyhedral space; intrinsic isometry; polyhedral space
UR - http://eudml.org/doc/276005
ER -

References

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  1. [1] U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on Rm, J. Geom., 16 (1981), no. 2, 187-193.  Zbl0467.51020
  2. [2] M. Bridson A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag Berlin Heidelberg, 1999.  Zbl0988.53001
  3. [3] D. Burago, Y. Burago, S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 2001.  Zbl0981.51016
  4. [4] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser, 2001.  
  5. [5] S. Krat, Approximation problems in length geometry, PhD thesis, The Pennsylvania State University, State College, PA, 2004.  
  6. [6] E. Le Donne, Lipschitz and path isometric embeddings of metric spaces, Geom. Dedicata, 166 (2012), 47-66.  Zbl1281.30041
  7. [7] B. Minemyer, Isometric Embeddings of Polyhedra into Euclidean Space, J. Topol. Anal. (in press), DOI:10.1142/S179352531550020X. [Crossref] Zbl1329.53062
  8. [8] B. Minemyer, Isometric Embeddings of Polyhedra, PhD thesis, The State University of New York at Binghamton, Binghamton, NY, 2013.  Zbl1329.53062
  9. [9] G. Moussong, Hyperbolic Coxeter Groups, PhD thesis, The Ohio State University, Columbus, OH, 1988.  
  10. [10] J. Nash, C1 Isometric Imbeddings, Ann. of Math. (2), 60 (1954), 383-396.  Zbl0058.37703
  11. [11] J. Nash, The Imbedding Problem for Riemannian Manifolds, Ann. of Math. (2), 63 (1956), 20-63.  Zbl0070.38603
  12. [12] A. Petrunin, On Intrinsic Isometries to Euclidean Space, St. Petersburg Math. J., 22 (2011), 803-812.  Zbl1225.53041
  13. [13] H. Whitney, Geometric integration theory, Princeton University Press, 1957.  Zbl0083.28204
  14. [14] V. A. Zalgaller, Isometric imbedding of polyhedra, Dokl. Akad. Nauk (in Russian), 123 (1958), 599-601.  Zbl0094.36004

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