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Transactions of the American Mathematical Society

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Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces


Authors: Gen Nakamura and Yoshiaki Maeda
Journal: Trans. Amer. Math. Soc. 313 (1989), 1-51
MSC: Primary 58G15; Secondary 35L99, 53C42
DOI: https://doi.org/10.1090/S0002-9947-1989-0992597-8
MathSciNet review: 992597
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Abstract: Local smooth isometric embedding problems of low dimensional Riemannian manifolds into Euclidean spaces are studied. Namely, we prove the existence of a local smooth isometric embedding of a smooth Riemannian $ 3$-manifold with nonvanishing curvature into Euclidean $ 6$-space. For proving this, we give a local solvability theorem for a system of a nonlinear PDE of real principal type.

To obtain the local solvability theorem, we need a tame estimate for the linearized equation corresponding to the given PDE, which is presented by two methods. The first is based on the result of Duistermaat-Hörmander which constructed the exact right inverse for linear PDEs of real principal type by using Fourier integral operators. The second method uses more various properties of Fourier integral operators given by Kumano-go, which seems to be a simpler proof than the above.


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  • [BGY] R. Bryant, P. A. Griffiths and D. Yang, Characteristics and existence of isometric embeddings, Duke Math. J. 3 (1983), 893-994. MR 726313 (85d:53027)
  • [D] J. J. Duistermaat, Fourier integral operators, Courant Institute of Math. Sci., New York Univ., 1973. MR 0451313 (56:9600)
  • [DH] J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), 183-269. MR 0388464 (52:9300)
  • [DY] D. DeTurck and D. Yang, Local existence of smooth metrics with prescribed curvature, Nonlinear Problems in Geometry, (D. DeTurck, ed.), Contemp. Math., vol. 51, Amer. Math. Soc., Providence, R.I., 1985, pp. 37-43. MR 848931 (87j:53059)
  • [E] Ju. V. Egorov, Canonical transformations and pseudodifferential operators, Trans. Moscow Math. Soc. 24 (1971), 1-28. MR 0361929 (50:14371)
  • [GY] J. B. Goodman and D. Yang, Local solvability of nonlinear partial differential equations of real principal type (to appear).
  • [H] L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), 79-183. MR 0388463 (52:9299)
  • [H1] -, The analysis of linear partial differential operators IV, Grundlehren Math. Wiss., vol. 275, Springer-Verlag, 1985. MR 781537 (87d:35002b)
  • [J] H. Jacobowitz, Local isometric embeddings, Ann. of Math. Stud., no. 102, Princeton Univ. Press, 1982. MR 645749 (83g:53022)
  • [K] H. Kumano-go, Pseudo-differential operators, MIT Press, Cambridge, Mass., 1981.
  • [L] C. S. Lin, The local isometric embedding $ {{\mathbf{R}}^3}$ of two dimensional Riemannian manifolds with Gaussian curvature changing sign cleanly, Ph.D. dissertation, Courant Institute, 1983.
  • [M] S. Mizohata, The theory of partial differential equations, Cambridge Univ. Press, Cambridge, 1973. MR 0599580 (58:29033)
  • [OMY1] H. Omori, Y. Maeda and A. Yoshioka, On regular Fréchet-Lie groups. I, Some differential geometrical expressions of Fourier-integral operators on a Riemannian manifold, Tolkyo J. Math. 3 (1980), 353-390. MR 605098 (82b:58089)
  • [OMY2] -, On regular Fréchet-Lie groups. II, Composition rules of Fourier-integral operators on a Riemannian manifold, Tokyo J. Math. 4 (1981), 221-253. MR 646038 (83g:58066)
  • [OMYK3] H. Omori, Y. Maeda, A. Yoshioka and O. Kobayashi, On regular Fréchet-Lie groups. III, A second cohomology class related to the Lie algebra of pseudo-differential operators of order one, Tokyo J. Math. 4 (1981), 255-277. MR 646039 (83g:58067)
  • [OMYK4] -, On regular Fréchet-Lie groups. IV, Definition and fundamental theorems, Tokyo J. Math. 5 (1982), 365-398. MR 688131 (84h:22044)
  • [OMYK5] -, On regular Fréchet-Lie groups. V, Several basic properties, Tokyo J. Math. 6 (1983), 39-64. MR 707838 (85h:22024)
  • [OMYK6] -, On regular Fréchet-Lie groups. VII, The group generated by pseudo-differential operators of negative order, Tokyo J. Math. 7 (1984), 315-336. MR 776942 (87b:22038)
  • [OMYK7] -, On regular Fréchet-Lie groups. VIII, Primordial operators and Fourier integral operators, Tokyo J. Math. 8 (1985), 1-47. MR 800075 (87b:22039)
  • [S] M. F. Sergeraert, Une généralisation du théorème des fonctions implicites de Nash, C. R. Acad. Sci. Paris 270A (1970), 861-863. MR 0259699 (41:4332)
  • [T] J. F. Treves, Linear partial differential equations with constant coefficients, Gordon and Breach, New York, 1966. MR 0224958 (37:557)
  • [Y] S. T. Yau, Seminar on differential geometry, Ann. of Math. Stud., no. 102, Princeton Univ. Press, 1982.

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DOI: https://doi.org/10.1090/S0002-9947-1989-0992597-8
Article copyright: © Copyright 1989 American Mathematical Society

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