Bulletin of the American Mathematical Society

MathSciNet review: 1567709
Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: Phillip A. Griffiths and Gary R. Jensen
Title: Differential systems and isometric embeddings
Additional book information: Annals of Mathematics Studies, vol. 114, Princeton University Press, Princeton, N. J., 1987, xii+225 pp., $35.00 (cloth). $15.00 (paper). ISBN 0-691-08429-7.

Takao Akahori, A new approach to the local embedding theorem of CR-structures for $n\geq 4$ (the local solvability for the operator $\overline \partial _b$ in the abstract sense), Mem. Amer. Math. Soc. 67 (1987), no. 366, xvi+257. MR 888499, DOI 10.1090/memo/0366

Eric Berger, Robert Bryant, and Phillip Griffiths, The Gauss equations and rigidity of isometric embeddings, Duke Math. J. 50 (1983), no. 3, 803–892. MR 714831, DOI 10.1215/S0012-7094-83-05039-1

Robert L. Bryant, Phillip A. Griffiths, and Deane Yang, Characteristics and existence of isometric embeddings, Duke Math. J. 50 (1983), no. 4, 893–994. MR 726313, DOI 10.1215/S0012-7094-83-05040-8

[Br] R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths and D. Yang, Essays on exterior differential systems (in preparation).

[Bu] C. Burstin, Ein Beitragzum Problem der Einbettung der Riemannschen Raume in Euklidishen Raume, Math. Sb. 38 (1931), 74-85.

Shiing-Shen Chern and Claude Chevalley, Obituary: Elie Cartan and his mathematical work, Bull. Amer. Math. Soc. 58 (1952), 217–250. MR 46972, DOI 10.1090/S0002-9904-1952-09588-4

[Ea] M. Eastwood, The Hill-Penrose-Sparling C.R.-folds, Twistor Newsletter 18 (1984), 16.

[Go] E. Goursat, Leçons sur le probleme de Pfaff, Gauthier-Villars, Paris, 1922.

Robert 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

Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222. MR 656198, DOI 10.1090/S0273-0979-1982-15004-2

[Ho] L. Hormander, Implicit function theorems, mimeographed notes of lectures given at Stanford University, 1977.

H. Jacobowitz, Extending isometric embeddings, J. Differential Geometry 9 (1974), 291–307. MR 377773

Howard Jacobowitz, Homogeneous solvability and CR structures, Notas de Curso [Course Notes], vol. 25, Universidade Federal de Pernambuco, Departamento de Matemática, Recife, 1988. MR 934571

Howard Jacobowitz and François Trèves, Nonrealizable CR structures, Invent. Math. 66 (1982), no. 2, 231–249. MR 656622, DOI 10.1007/BF01389393

Howard Jacobowitz and François Trèves, Aberrant CR structures, Hokkaido Math. J. 12 (1983), no. 3, 276–292. MR 719968, DOI 10.14492/hokmj/1470081006

Masatake Kuranishi, Strongly pseudoconvex CR structures over small balls. I. An a priori estimate, Ann. of Math. (2) 115 (1982), no. 3, 451–500. MR 657236, DOI 10.2307/2007010

[Li] Chang-Shou Lin, The local isometric embedding problem in R, Thesis, NYU, New York 1983.

Jürgen Moser, A rapidly convergent iteration method and non-linear differential equations. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 499–535. MR 206461

John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63. MR 75639, DOI 10.2307/1969989

L. Nirenberg, A certain problem of Hans Lewy, Uspehi Mat. Nauk 29 (1974), no. 2(176), 241–251 (Russian). Translated from the English by Ju. V. Egorov; Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. MR 0492752

J. Schwartz, On Nash’s implicit functional theorem, Comm. Pure Appl. Math. 13 (1960), 509–530. MR 114144, DOI 10.1002/cpa.3160130311

Francis Sergeraert, Une généralisation du théorème des fonctions implicites de Nash, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A861–A863 (French). MR 259699

[Sp] M. Spivak, Differential geometry, Volume 5, Publish or Perish, Boston, 1975.

[We] J. Weingarten, Über die Theorie der aufeinander abwickelbaren Oberflachen, Berlin, 1884.

[Wy] H. Weyl, Cartan on groups and differential geometry, Bull. Amer. Math. Soc. 44 (1938), 598-601.

[Ya] D. Yang, Involutive hyperbolic differential systems, thesis, Harvard Univ., Cambridge, Mass., 1982.

[Ak] T. Akahori, A new approach to the local embedding theorem of CR-structures of n ≥ 4... Mem. Amer. Math. Soc., no. 336, Amer. Math. Soc., Providence, R. I., 1987. MR 0888499




[BBG] E. Berger, R. Bryant and P. Griffiths, The Gauss equations and rigidity of isometric embeddings, Duke Math. J. 50 (1983), 800-892. MR 714831




[BGY] R. Bryant, P. Griffiths and D. Yang, Characteristics and existence of isometric embeddings, Duke Math. J. 50 (1983), 893-994. MR 726313




[Br] R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths and D. Yang, Essays on exterior differential systems (in preparation).




[Bu] C. Burstin, Ein Beitragzum Problem der Einbettung der Riemannschen Raume in Euklidishen Raume, Math. Sb. 38 (1931), 74-85.




[CC] S. S. Chern and C. Chevalley, Elie Cartan and his mathematical work, Bull. Amer. Math. Soc. 58 (1952), 217-250. MR 46972




[Ea] M. Eastwood, The Hill-Penrose-Sparling C.R.-folds, Twistor Newsletter 18 (1984), 16.




[Go] E. Goursat, Leçons sur le probleme de Pfaff, Gauthier-Villars, Paris, 1922.




[Gr] R. Greene, Isometric embeddings of Riemannian and pseudo-Riemannian manifolds, Mem. Amer. Math. Soc. no. 97, Amer. Math. Soc., Providence, R. I., 1970. MR 262980




[Ha] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N. S. ) 7 (1982), 65-222. MR 656198




[Ho] L. Hormander, Implicit function theorems, mimeographed notes of lectures given at Stanford University, 1977.




[Ja1] H. Jacobowitz, Extending isometric embeddings, J. Differential Geometry 9 (1974), 291-307. MR 377773




[Ja2] H. Jacobowitz, Homogeneous solvability and CR structures, Notas de Curso, Univerśidade Pederal de Pernambuco, Recife, Brazil, 1988. MR 934571




[JT1] H. Jacobowitz and F. Treves, Nonrealizable CR structures, Invent. Math. 66 (1982), 231-249. MR 656622




[JT2] H. Jacobowitz and F. Treves, Abberant CR structures, Hokkaido Math. J. 12 (1983), 276-292. MR 719968




[Ku] M. Kuranishi, Strongly pseudoconvex CR structures over small balls, Part III. An embedding theorem. Ann. of Math. (2) 116 (1982), 249-330. MR 672837




[Li] Chang-Shou Lin, The local isometric embedding problem in R, Thesis, NYU, New York 1983.




[Mo] J. Moser, A rapidly converging iteration method and non-linear differential equations, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 499-535. MR 206461




[Na] J. Nash, The embedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-64. MR 75639




[Ni] L. Nirenberg, On a problem of Hans Lewy, Uspekhi Mat. Nauk 292 (176) (1974), 241-251. MR 492752




[Sc] J. Schwartz, On Nash's implicit functional theorem, Comm. Pure Appl. Math. 13 (1960), 509-530. MR 114144




[Se] F. Sergeraert, Une generalisation du théorème des fonctiones implicites de Nash, C. R. Acad. Sci. Paris Ser. A 270 (1970), 861-863. MR 259699




[Sp] M. Spivak, Differential geometry, Volume 5, Publish or Perish, Boston, 1975.




[We] J. Weingarten, Über die Theorie der aufeinander abwickelbaren Oberflachen, Berlin, 1884.




[Wy] H. Weyl, Cartan on groups and differential geometry, Bull. Amer. Math. Soc. 44 (1938), 598-601.




[Ya] D. Yang, Involutive hyperbolic differential systems, thesis, Harvard Univ., Cambridge, Mass., 1982.