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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
\Bibitem{GroRok70}
\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
\mathnet{http://mi.mathnet.ru/umn5402}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=290390}
\zmath{https://zbmath.org/?q=an:0202.21004|0222.53053}
\transl
\jour Russian Math. Surveys
\yr 1970
\vol 25
\issue 5
\pages 1--57
\crossref{https://doi.org/10.1070/RM1970v025n05ABEH003801}


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    This publication is cited in the following articles:
    1. M. L. Gromov, Ya. M. Eliashberg, “Removal of singularities of smooth mappings”, Math. USSR-Izv., 5:3 (1971), 615–639  mathnet  crossref  mathscinet  zmath
    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
    4. M. L. Gromov, “Smoothing and inversion of differential operators”, Math. USSR-Sb., 17:3 (1972), 381–435  mathnet  crossref  mathscinet  zmath
    5. È. G. Poznyak, “Isometric immersions of two-dimensional Riemannian metrics in euclidean space”, Russian Math. Surveys, 28:4 (1973), 47–77  mathnet  crossref  mathscinet  zmath
    6. M. L. Gromov, “Convex integration of differential relations. I”, Math. USSR-Izv., 7:2 (1973), 329–343  mathnet  crossref  mathscinet  zmath
    7. João Lucas Barbosa, Manfredo do Carmo, “A proof of a general isoperimetric inequality for surfaces”, Math Z, 162:3 (1978), 245  crossref  mathscinet  zmath
    8. Howard Jacobowitz, “Curvature operators on the exterior algebra1†”, Linear and Multilinear Algebra, 7:2 (1979), 93  crossref
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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
    11. V. A. Aleksandrov, “Immersions of locally euclidean and conformally flat metrics”, Math. USSR-Sb., 73:2 (1992), 467–478  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. Nobumitsu Nakauchi, “Homomorphism between homotopy groups induced by elements of the sobolev space L1,p (M, N)”, manuscripta math, 78:1 (1993), 1  crossref  mathscinet  zmath  isi
    13. Thomas T. Burwick, “All-order quantum gravity in two dimensions”, Nuclear Physics B, 418:1-2 (1994), 257  crossref
    14. Sun-Yung A. Chang, Jie Qing, “The Zeta Functional Determinants on Manifolds with Boundary”, Journal of Functional Analysis, 147:2 (1997), 363  crossref
    15. István Ozsváth, Engelbert L. Schücking, “The world viewed from outside”, Journal of Geometry and Physics, 24:4 (1998), 303  crossref
    16. I. G. Tsar'kov, E. T. Shavgulidze, “Smooth Isometric Immersions into the Infinite-Dimensional Sphere”, Funct. Anal. Appl., 33:3 (1999), 239–240  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. S. C. Venkataramani, T. A. Witten, E. M. Kramer, R. P. Geroch, “Limitations on the smooth confinement of an unstretchable manifold”, J Math Phys (N Y ), 41:7 (2000), 5107  crossref  mathscinet  zmath  adsnasa  isi
    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
    19. YURI KIFER, “STOCHASTIC VERSIONS OF ANOSOV'S AND NEISTADT'S THEOREMS ON AVERAGING”, Stoch. Dyn, 01:01 (2001), 1  crossref
    20. Hui Xia He, Zi Zhou Tang, “An Isometric Embedding of Möbius Band with Positive Gaussian Curvature”, Acta Math Sinica, 20:6 (2004), 961  crossref  mathscinet  zmath
    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
    23. Emil Saucan, Eli Appleboim, Yehoshua Y. Zeevi, “Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds”, J Math Imaging Vis, 30:1 (2008), 105  crossref  mathscinet  isi
    24. Hiroyuki Chihara, Eiji Onodera, “A third order dispersive flow for closed curves into almost Hermitian manifolds”, Journal of Functional Analysis, 257:2 (2009), 388  crossref
    25. Jost-Hinrich Eschenburg, B.S. Kruglikov, Vladimir S. Matveev, Renato Tribuzy, “Compatibility of Gauß maps with metrics”, Differential Geometry and its Applications, 28:2 (2010), 228  crossref
    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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