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Uspekhi Mat. Nauk, 1970, Volume 25, Issue 5(155), Pages 3–62 (Mi umn5402)  

This article is cited in 29 scientific papers (total in 30 papers)

Embeddings and immersions in Riemannian geometry

M. L. Gromov, V. A. Rokhlin

Abstract: This article is a significantly expanded version of a paper read by one of the authors to the Moscow Mathematical Society [18]. It consists of two chapters and eleven appendices. The first chapter contains a survey of known results, as a rule with precise statements and references, but without full proofs. In the second chapter, the fundamental embedding theorems are set out in detail, among them some new results concerning a bound for the smallest dimension of a euclidean space in which any compact riemannian manifold of given dimension can be embedded, and also to the corresponding local problem. In Appendices 1, 3, 4, 5, 7, 8 and 9, proofs of miscellaneous propositions in the survey are given. Appendices 2 and 6 are needed for other appendices, but are interesting in their own right. In Appendix 10 a more general embedding problem is considered. Appendix 11 has a bearing on Chapter 2 and is of an analytic nature.
We wish to thank I. A. Bakel'man, A. L. Verner, Yu. A. Volkov, S. P. Geisberg, V. L. Eidlin and Ya. M. Eliashberg for their help in the preparation of the article. We are especially grateful to Yu. D. Burago who at our request modified one of his inequalities to suit our requirements (Appendix 2).

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English version:
Russian Mathematical Surveys, 1970, 25:5, 1–57

Bibliographic databases:

UDC: 513.813
MSC: 58B20, 57R40, 57N35
Received: 13.05.1970

Citation: M. L. Gromov, V. A. Rokhlin, “Embeddings and immersions in Riemannian geometry”, Uspekhi Mat. Nauk, 25:5(155) (1970), 3–62; Russian Math. Surveys, 25:5 (1970), 1–57

Citation in format AMSBIB
\by M.~L.~Gromov, V.~A.~Rokhlin
\paper Embeddings and immersions in Riemannian geometry
\jour Uspekhi Mat. Nauk
\yr 1970
\vol 25
\issue 5(155)
\pages 3--62
\jour Russian Math. Surveys
\yr 1970
\vol 25
\issue 5
\pages 1--57

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    This publication is cited in the following articles:
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    2. D. Nesh, “Problema vlozhenii dlya rimanovykh mnogoobrazii”, UMN, 26:4(160) (1971), 173–216  mathnet  mathscinet  zmath
    3. D. Nesh, “Analitichnost reshenii zadach o neyavnoi funktsii s analiticheskimi iskhodnymi dannymi”, UMN, 26:4(160) (1971), 217–226  mathnet  mathscinet  zmath
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    10. Kevin M. Short, “New geometric formulation of string theory”, J Math Phys (N Y ), 32:1 (1991), 280  crossref  mathscinet  zmath  isi
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    13. Thomas T. Burwick, “All-order quantum gravity in two dimensions”, Nuclear Physics B, 418:1-2 (1994), 257  crossref
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    18. A. A. Borisenko, “Isometric immersions of space forms into Riemannian and pseudo-Riemannian spaces of constant curvature”, Russian Math. Surveys, 56:3 (2001), 425–497  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
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    21. ZIZHOU TANG, “SOME EXISTENCE AND NONEXISTENCE RESULTS OF ISOMETRIC IMMERSIONS OF Riemannian MANIFOLDS”, Commun. Contemp. Math, 06:06 (2004), 867  crossref
    22. Hwajeong Kim, “A variational approach to the regularity of minimal surfaces of annulus type in Riemannian manifolds”, Differential Geometry and its Applications, 25:5 (2007), 466  crossref
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    26. Mirandola H. Vitorio F., “Global Isometric Embeddings of Multiple Warped Product Metrics Into Quadrics”, 38, no. 1, 2015, 119–134  isi
    27. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175  mathnet  crossref  elib
    28. Gromov M., “Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems”, Bull. Amer. Math. Soc., 54:2 (2017), 173–245  crossref  mathscinet  zmath  isi  scopus
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