Local smooth isometric embeddings of lowdimensional Riemannian manifolds into Euclidean spaces
Authors:
Gen Nakamura and Yoshiaki Maeda
Journal:
Trans. Amer. Math. Soc. 313 (1989), 151
MSC:
Primary 58G15; Secondary 35L99, 53C42
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 manifold with nonvanishing curvature into Euclidean 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 DuistermaatHörmander 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 Kumanogo, which seems to be a simpler proof than the above.
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S. T. Yau, Seminar on differential geometry, Ann. of Math. Stud., no. 102, Princeton Univ. Press, 1982.
 [BGY]
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 [D]
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 [DH]
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 [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. 3743. MR 848931 (87j:53059)
 [E]
 Ju. V. Egorov, Canonical transformations and pseudodifferential operators, Trans. Moscow Math. Soc. 24 (1971), 128. 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), 79183. MR 0388463 (52:9299)
 [H1]
 , The analysis of linear partial differential operators IV, Grundlehren Math. Wiss., vol. 275, SpringerVerlag, 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. Kumanogo, Pseudodifferential operators, MIT Press, Cambridge, Mass., 1981.
 [L]
 C. S. Lin, The local isometric embedding 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 FréchetLie groups. I, Some differential geometrical expressions of Fourierintegral operators on a Riemannian manifold, Tolkyo J. Math. 3 (1980), 353390. MR 605098 (82b:58089)
 [OMY2]
 , On regular FréchetLie groups. II, Composition rules of Fourierintegral operators on a Riemannian manifold, Tokyo J. Math. 4 (1981), 221253. MR 646038 (83g:58066)
 [OMYK3]
 H. Omori, Y. Maeda, A. Yoshioka and O. Kobayashi, On regular FréchetLie groups. III, A second cohomology class related to the Lie algebra of pseudodifferential operators of order one, Tokyo J. Math. 4 (1981), 255277. MR 646039 (83g:58067)
 [OMYK4]
 , On regular FréchetLie groups. IV, Definition and fundamental theorems, Tokyo J. Math. 5 (1982), 365398. MR 688131 (84h:22044)
 [OMYK5]
 , On regular FréchetLie groups. V, Several basic properties, Tokyo J. Math. 6 (1983), 3964. MR 707838 (85h:22024)
 [OMYK6]
 , On regular FréchetLie groups. VII, The group generated by pseudodifferential operators of negative order, Tokyo J. Math. 7 (1984), 315336. MR 776942 (87b:22038)
 [OMYK7]
 , On regular FréchetLie groups. VIII, Primordial operators and Fourier integral operators, Tokyo J. Math. 8 (1985), 147. MR 800075 (87b:22039)
 [S]
 M. F. Sergeraert, Une généralisation du théorème des fonctions implicites de Nash, C. R. Acad. Sci. Paris 270A (1970), 861863. 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:
http://dx.doi.org/10.1090/S00029947198909925978
PII:
S 00029947(1989)09925978
Article copyright:
© Copyright 1989
American Mathematical Society
