Isometric Embeddings of Pro-Euclidean Spaces

• Autorzy: Barry Minemyer
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

differential geometry; discrete geometry; metric geometry; Euclidean polyhedra; polyhedralspace; intrinsic isometry

• Wydawca: De Gruyter Open
• Czasopismo: Analysis and Geometry in Metric Spaces
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2015-05-27

zaakceptowano

2015-10-09

online

2015-10-29

Twórcy

autor

Barry Minemyer

• Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

Bibliografia

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

Identyfikatory

DOI

10.1515/agms-2015-0019