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Isometric embedding of a compact Riemannian manifold into Euclidean space

Author: Howard Jacobowitz
Journal: Proc. Amer. Math. Soc. 40 (1973), 245-246
MSC: Primary 53C40
MathSciNet review: 0375173
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Abstract: An isometric immersion of an $ n$-dimensional compact Riemannian manifold with sectional curvature always less than $ {\lambda ^{ - 2}}$ into Euclidean space of dimension $ 2n - 1$ can never be contained in a ball of radius $ \lambda $. This generalizes and includes results of Tompkins and Chern and Kuiper.

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  • [1] S. S. Chern and N. H. Kuiper, Some theorems on the isometric embedding of compact Riemann manifolds in euclidean space, Ann. of Math. (2) 56 (1952), 422-430. MR 14, 408. MR 0050962 (14:408e)
  • [2] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Appl. Math., no. 15, Interscience, New York, 1969. MR 38 #6501. MR 0238225 (38:6501)
  • [3] T. Otsuki, On the existence of solutions of a system of quadratic equations and its geometrical application, Proc. Japan Acad. 29 (1953), 99-100. MR 15, 647. MR 0060281 (15:647e)
  • [4] C. Tompkins, Isometric embedding of flat manifolds in Euclidean space, Duke Math. J. 5 (1939), 58-61. MR 1546106

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Keywords: Isometric embedding, sectional curvature, Tompkin counter-example
Article copyright: © Copyright 1973 American Mathematical Society

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