Formal integrability of systems of partial differential equations

  • J. Gasqui
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 226)


We review some basic definitions and results of the formal theory of overdetermined systems of partial differential equations. We have tried to present them the most simply possible, in a language for non-experts. As illustration, we treat a familiar example for physicists : the Einstein equations.


Vector Bundle Formal Solution Einstein Equation Ricci Curvature Linear Differential Operator 
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The reader can consult: For linear equations

  1. H. GOLDSCHMIDT, Existence theorems for analytic linear partial differential equations, Ann. of Math. (1967), 246–270.Google Scholar
  2. D. C. SPENCER, Overdetermined systems of linear partial differential equations, Bull. A.M.S. 75 (1969), 179–239.Google Scholar

For non-linear equation

  1. H. GOLDSCHMIDT, Integrability criteria for systems of non linear partial differential equations, J. Differential Geometry, 1 (1967), 269–307.Google Scholar

For the proof of existence of local solutions in the analytic case

  1. B. MALGRANGE, Equations de Lie II, J. Differential Geometry, 7 (1972), 117–141.Google Scholar

For examples

  1. M. DLBOIS-VIOLETTE, The theory of overdetermined linear systems and its applications to non-linear field equations, (Lectures at the Banach Center, Warsaw, October 83).Google Scholar
  2. J. GASQUI, Sur l'existence locale d'immersions ä courbure scalaire donnée, Math. Annalen, 219 (1976), 123–126.CrossRefGoogle Scholar
  3. J. GASQUI, Sur l'existence locale de certaines métriques riemanniennes plates, Duke Math. J. 46 (1979), 109–118.CrossRefGoogle Scholar

For general applications in Geometry, the introduction of

  1. J. GASQUI-H. GOLDSCHMIDT, Déformations.infinitésimales des structures conformes plates, Progress in Math., Birkhäuser (1984).Google Scholar

For Einstein equations or Ricci curvature

  1. M. DUBOIS-VIOLETTE, Remarks on the local structure of Yang-Mills and Einstein equations, Phys. Lett. 131 B (1983) 323–326.Google Scholar
  2. J. GASQUI, Sur la résolubilité locale des equations d'Einstein, Compositio Math. 47 (1982), 43–69.Google Scholar
  3. D. DE Turck, Existence of Metrics with prescribed Ricci curvature: Local theory, Inventiones Math. 65 (1983), 327–349.Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • J. Gasqui
    • 1
  1. 1.Laboratoire de MathématiquesInstitut FourierSt Martin D'HeresFrance

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