Connections with prescribed curvature
Annales de l'Institut Fourier, Volume 37 (1987) no. 4, p. 29-44

We discuss the problem of prescribing the curvature of a connection on a principal bundle whose base manifold is three-dimensional. In particular, we consider the local question: Given a curvature form F, when does there exist locally a connection A such that F is the curvature of A ? When the structure group of the bundle is semisimple, a finite number of nonlinear identities arise as necessary conditions for local solvability of the curvature equation. We conjecture that these conditions are also generically sufficient, and we prove this for bundles whose structure group is of low rank. Nilpotent structure groups are also discussed.

Nous considérons le problème émanant de la prescription de la courbure d’une connexion sur un fibré principal dont la base est de dimension trois. En particulier, étant donné une forme de courbure F, quand existe-t-il localement une connexion A dont la courbure soit F ? Lorsque le groupe de structure du fibré est semi-simple, certaines identités non-linéaires, en nombre fini, apparaissent comme conditions nécessaires pour la résolution de l’équation de courbure. Nous conjecturons que ces conditions sont presque toujours suffisantes; nous donnons une preuve de ceci pour les fibrés dont le groupe de structure est de rang inférieur ou égal à trois. Nous étudions également des fibrés dont le groupe de structure est nilpotent.

@article{AIF_1987__37_4_29_0,
     author = {Deturck, Dennis and Talvacchia, Janet},
     title = {Connections with prescribed curvature},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {37},
     number = {4},
     year = {1987},
     pages = {29-44},
     doi = {10.5802/aif.1109},
     zbl = {0627.53027},
     mrnumber = {89d:53058},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1987__37_4_29_0}
}
Deturck, Dennis; Talvacchia, Janet. Connections with prescribed curvature. Annales de l'Institut Fourier, Volume 37 (1987) no. 4, pp. 29-44. doi : 10.5802/aif.1109. http://www.numdam.org/item/AIF_1987__37_4_29_0/

[1] A. Asada, Non-abelian Poincaré lemma, preprint, 1986. | MR 88d:58005 | Zbl 0607.58002

[2] A. Besse, Einstein manifolds, Springer-Verlag, Berlin, 1987. | MR 88f:53087 | Zbl 0613.53001

[3] R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt and P. Griffiths, Exterior differential systems, to appear. | Zbl 0726.58002

[4] E. Cartan, Les systèmes différentiels extérieurs et leurs applications géométriques, Paris, Hermann, 1945. | MR 7,520d | Zbl 0063.00734

[5] D. Deturck, Existence of metrics with prescribed Ricci curvature : Local theory, Inventiones Math., 65 (1981), 179-207. | MR 83b:53019 | Zbl 0489.53014

[6] D. Deturck and D. Yang, Local existence of smooth metrics with prescribed curvature, Contemporary Math., 51 (1986), 37-43. | MR 87j:53059 | Zbl 0587.53040

[7] J. Gasqui, Sur la résolubilité locale des équations d'Einstein, Compositio Math., 47 (1982), 43-69. | Numdam | MR 84f:58115 | Zbl 0515.53015

[8] J. Gasqui, Sur l'existence locale d'immersions à courbure scalaire donnée, Math. Ann., 219 (1976), 123-126. | MR 52 #12013 | Zbl 0303.53025

[9] H. Goldschmidt, Existence theorems for analytic linear partial differential equations, Annals of Math., 86 (1967), 246-270. | MR 36 #2933 | Zbl 0154.35103

[10] H. Goldschmidt, Integrability criteria for systems of non-linear partial differential equations, J. Diff. Geom., 1 (1967), 269-307. | MR 37 #1746 | Zbl 0159.14101

[11] V. Guillemin and S. Sternberg, An algebraic model of transitive differential geometry, Bulletin Amer. Math. Soc., 70 (1967), 16-47. | MR 30 #533 | Zbl 0121.38801

[12] M. Kuranishi, Lectures on involutive systems of partial differential equations, Lecture notes, Soc. Math. Sao Paulo, 1967. | Zbl 0163.12001

[13] B. Malgrange, Équations de Lie II, J. Diff. Geom., 7 (1972), 117-141. | MR 48 #5128 | Zbl 0264.58009

[14] J. Talvacchia, Ph. D Thesis, U. of Pennsylvania.

[15] S. P. Tsarev, Which 2-forms are locally, curvature forms? Funct. Anal. and Applications, 16 (1982), 90-91 (Russian, English Translation, 16 (1982), 235-237). | MR 84e:53043 | Zbl 0516.53014

[16] D. Zhelobenko, Compact Lie groups and their representations, A.M.S. Translations of Math. Monographs, 40 (1973). | Zbl 0272.22006