Isometric embedding of a compact Riemannian manifold into Euclidean space
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- by Howard Jacobowitz PDF
- Proc. Amer. Math. Soc. 40 (1973), 245-246 Request permission
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.References
- Shiing-shen Chern and Nicolaas H. Kuiper, Some theorems on the isometric imbedding of compact Riemann manifolds in euclidean space, Ann. of Math. (2) 56 (1952), 422–430. MR 50962, DOI 10.2307/1969650
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
- Tominosuke Otsuki, On the existence of solutions of a system of quadratic equations and its geometrical application, Proc. Japan Acad. 29 (1953), 99–100. MR 60281
- C. Tompkins, Isometric embedding of flat manifolds in Euclidean space, Duke Math. J. 5 (1939), no. 1, 58–61. MR 1546106, DOI 10.1215/S0012-7094-39-00507-7
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 40 (1973), 245-246
- MSC: Primary 53C40
- DOI: https://doi.org/10.1090/S0002-9939-1973-0375173-3
- MathSciNet review: 0375173