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ISSN 1088-6850(e) ISSN 0002-9947(p)

Relative isometric embeddings of Riemannian manifolds

Author(s): Mohammad Ghomi; Robert E. Greene
Journal: Trans. Amer. Math. Soc. 363 (2011), 63-73.
MSC (2010): Primary 53C42, 53A07
Posted: August 16, 2010
MathSciNet review: 2719671
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Abstract: We prove the existence of isometric embeddings, and $C^\infty$ approximate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed values in a neighborhood of a point.

References:

1.: S. Alexander, M. Ghomi, and J. Wong.
Topology of Riemannian submanifolds with prescribed boundary, Duke Math. J., 152 (2010), 533-565.
2.: B. Andrews.
Notes on the isometric embedding problem and the Nash-Moser implicit function theorem.
In Surveys in analysis and operator theory (Canberra, 2001), volume 40 of Proc. Centre Math. Appl. Austral. Nat. Univ., pages 157-208. Austral. Nat. Univ., Canberra, 2002. MR 1953483 (2004c:53075)
3.: Y. Eliashberg and N. Mishachev.
Introduction to the -principle, volume 48 of Graduate Studies in Mathematics.
American Mathematical Society, Providence, RI, 2002. MR 1909245 (2003g:53164)
4.: M. Ghomi.
The problem of optimal smoothing for convex functions.
Proc. Amer. Math. Soc., 130(8):2255-2259 (electronic), 2002. MR 1896406 (2004d:26013)
5.: R. E. Greene.
Isometric embeddings of Riemannian and pseudo-Riemannian manifolds.
Memoirs of the American Mathematical Society, No. 97. American Mathematical Society, Providence, R.I., 1970. MR 0262980 (41:7585)
6.: M. Gromov.
Partial differential relations.
Springer-Verlag, Berlin, 1986. MR 864505 (90a:58201)
7.: M. Günther.
Isometric embeddings of Riemannian manifolds.
In Proceedings of the International Congress of Mathematicians, Vols. I, II (Kyoto, 1990), pages 1137-1143, Math. Soc. Japan, Tokyo, 1991. MR 1159298 (93b:53049)
8.: M. Günther.
On the perturbation problem associated to isometric embeddings of Riemannian manifolds.
Ann. Global Anal. Geom., 7(1):69-77, 1989. MR 1029846 (91a:58023)
9.: Q. Han and J.-X. Hong.
Isometric embedding of Riemannian manifolds in Euclidean spaces, volume 130 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 2006. MR 2261749 (2008e:53055)
10.: N. J. Hicks.
Notes on differential geometry.
Van Nostrand Mathematical Studies, No. 3. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0179691 (31:3936)
11.: H. Jacobowitz.
Extending isometric embeddings.
J. Differential Geometry, 9:291-307, 1974. MR 0377773 (51:13942)
12.: E. Kreyszig.
Differential geometry.
Dover Publications Inc., New York, 1991.
Reprint of the 1963 edition. MR 1118149 (92d:53001)
13.: N. H. Kuiper.
On $C\sp 1$ -isometric imbeddings. I, II.
Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math., 17:545-556, 683-689, 1955. MR 0075640 (17:782c)
14.: J. Nash.
$C\sp 1$ isometric imbeddings.
Ann. of Math. (2), 60:383-396, 1954. MR 0065993 (16:515e)
15.: J. Nash.
The imbedding problem for Riemannian manifolds.
Ann. of Math. (2), 63:20-63, 1956. MR 0075639 (17:782b)
16.: B. O'Neill.
Elementary differential geometry.
Elsevier/Academic Press, Amsterdam, second edition, 2006. MR 2351345 (2008k:53001)
17.: A. V. Pogorelov.
Extrinsic geometry of convex surfaces.
American Mathematical Society, Providence, R.I., 1973.
Translations of Mathematical Monographs, Vol. 35. MR 0346714 (49:11439)
18.: I. K. Sabitov.
Local theory of bendings of surfaces.
In Geometry, III, volume 48 of Encyclopaedia Math. Sci., pages 179-256. Springer, Berlin, 1992. MR 1306736
19.: M. Spivak.
A comprehensive introduction to differential geometry. Vol. V.
Publish or Perish Inc., Wilmington, Del., second edition, 1979. MR 0532834 (82g:53003e)

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