References
[A] Atiyah, M.: Elliptic Operators and Compact Groups. Lectures Notes in Math., vol.401. Berlin, Heidelberg, New York: Springer 1974
[B] Bourguignon, J.-P.: Ricci Curvature and Einstein Metrics. Global Differential Geometry/Global Analysis Proceedings, Berlin, Nov. 1979. Lecture Notes in Math., vol. 838, p. 42–63. Berlin, Heidelberg, New York, Springer 1981
[Be] Berger, M.: Quelques formules de variation pour une structure Riemannienne, Ann. Sci. Ecole Norm. Sup., 4e série, t.3, 285–294 (1970)
[BE] Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Differential Geometry3, 379–392 (1969)
[BJS] Bers, L., John, F., Schechter, M.: Partial Differential Equations. Wiley-Interscience, 1964
[C] Cartan, E.: Les systèmes différentiels extérieurs et leurs applications géometriques. Paris: Hermann 1945
[D1] DeTurck, D.: The Equation of Prescribed Ricci Curvature. Bull. Amer. Math. Soc.3, 701–704 (1980)
[D2] DeTurck, D.: Metrics with prescribed Ricci curvature. Proceedings of IAS Differential Geometry seminar, 1979–80, to appear in Annals of Math. Study 102
[DK] DeTurck, D., Kazdan, J.: Some regularity theorems in Riemannian geometry. To appear in Ann. Sci. Ecole Norm. Sup.
[G1] Gasqui, J.: Connexions à courbure de Ricci donnée, Math. Z.,168, 167–179 (1979)
[G2] Gasqui, J.: Sur l'existence locale d'immersions à courbure scalaire donnée. Math. Annalen,241, 283–288 (1979)
[G3] Gasqui, J.: Métriques Riemanniennes Ricci-intégrables et resolubilité locale des équations d'Einstein. Preprint, Universite de Grenoble I
[Gol] Goldschmidt, H.: Existence theorems for analytic linear partial differential equations. Ann. of Math.86, 246–270 (1967)
[Go2] Goldschmidt, H.: Integrability criteria for systems of non-linear partial differential equations. J. Differential Geometry1, 269–307 (1967)
[GS] Guillemin, V., Sternberg, S.: An algebraic model of transitive differential geometry. Bull. Amer. Math. Soc.70, 16–47 (1964)
[K1] Kazdan, J.: Another proof of Bianchi's identity in Riemannian geometry. Proc. Amer. Math. Soc.,81, 341–342 (1981)
[K2] Kazdan, J.: Gaussian and scalar curvature, an update. Proceedings of IAS Differential Geometry seminar, 1979–80, to appear in Annals of Math Study 102
[K3] Kazdan, J.: Partial Differential Equations. Lecture notes, University of Pennsylvania
[Ku] Kuranishi, M.: Lectures on Involutive Systems of Partial Differential Equations. Lecture notes, Soc. Math. Sao Paulo, 1967
[M1] Malgrange, B.: Sur l'intégrabilité des structures presque-complexes. Symposia Math. vol. II (INDAM, Rome, 1968), New York: Academic Press, London pp. 289–296, 1969
[M2] Malgrange, B.: Equations de Lie II. J. Differential Geometry7, 117–141 (1972)
[Mo] Moser, J.: A new technique for the construction of solutions of nonlinear differential equations. Proc. Natl. Acad. Sci.,47, 1824–1831 (1961)
[MR] Min-Oo, Ruh E.: Comparison theorems for compact symmetric spaces. Ann. Sci. Ecol. Norm. Sup. 4e serie, t.12, 335–353 (1979)
[N] Nirenberg, L.: Topics in Nonlinear Functional Analysis. Courant Institute Lecture Notes, NYU, 1974
[Na] Nash, J.: The imbedding of Riemannian manifolds. Ann. Math.63, 20–63 (1956)
[S] Singer, I.: Recent applications of index theory for elliptic operators. Proc. Symp. Pure Math., vol. XXIII (Partial Differential Equations), AMS, pp. 11–31, 1976
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The research for this paper was supported in part by the National Science Foundation
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DeTurck, D.M. Existence of metrics with prescribed Ricci curvature: Local theory. Invent Math 65, 179–207 (1981). https://doi.org/10.1007/BF01389010
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DOI: https://doi.org/10.1007/BF01389010