Lipschitz and path isometric embeddings of metric spaces

被引：17
作者：
Le Donne, Enrico
机构：
来源：
GEOMETRIAE DEDICATA | 2013年 / 166卷 / 01期
关键词：
Path isometry; Embedding; Sub-Riemannian manifold; Nash embedding theorem; Lipschitz embedding;
D O I：
10.1007/s10711-012-9785-2
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C-1 Embedding Theorem. For more general metric spaces the same result is false, e. g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Euclidean space via a Lipschitz map.
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页码：47 / 66
页数：20
相关论文
共 19 条
  • [1] [Anonymous], 2002, AM MATH SOC, DOI DOI 10.1090/SURV/091
  • [2] [Anonymous], 1999, Metric structures for Riemannian and nonRiemannian spaces
  • [3] [Anonymous], 1996, Progress in Mathematics
  • [4] [Anonymous], 2010, LECT NOTES SUBRIEMAN
  • [5] Buliga M., 2002, PREPRINT
  • [6] Burago D., 2001, A Course in Metric Geometry, DOI [10.1090/gsm/033, DOI 10.1090/GSM/033]
  • [7] NASH C-1-EMBEDDING THEOREM FOR CARNOT-CARATHEODORY METRICS
    DAMBRA, G
    [J]. DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 1995, 5 (02) : 105 - 119
  • [8] Federer H., 1959, Transactions of the American Mathematical Society, V93, P418, DOI [DOI 10.1090/S0002-9947-1959-0110078-1, 10.1090/S0002-9947-1959-0110078-1]
  • [9] Gromov M., 1986, ERGEBNISSE MATH IHRE, V9, DOI [10.1007/978-3-662-02267-2, DOI 10.1007/978-3-662-02267-2]
  • [10] Kuiper N.H., 1955, Nederl. Akad. Wetensch. Proc. Ser. A, P683
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