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

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.

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.

@article{AIF_1987__37_4_29_0,
     author = {Deturck, Dennis and Talvacchia, Janet},
     title = {Connections with prescribed curvature},
     journal = {Annales de l'Institut Fourier},
     pages = {29--44},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {4},
     year = {1987},
     doi = {10.5802/aif.1109},
     mrnumber = {89d:53058},
     zbl = {0627.53027},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1109/}
}
TY  - JOUR
AU  - Deturck, Dennis
AU  - Talvacchia, Janet
TI  - Connections with prescribed curvature
JO  - Annales de l'Institut Fourier
PY  - 1987
SP  - 29
EP  - 44
VL  - 37
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1109/
DO  - 10.5802/aif.1109
LA  - en
ID  - AIF_1987__37_4_29_0
ER  - 
%0 Journal Article
%A Deturck, Dennis
%A Talvacchia, Janet
%T Connections with prescribed curvature
%J Annales de l'Institut Fourier
%D 1987
%P 29-44
%V 37
%N 4
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1109/
%R 10.5802/aif.1109
%G en
%F AIF_1987__37_4_29_0
Deturck, Dennis; Talvacchia, Janet. Connections with prescribed curvature. Annales de l'Institut Fourier, Tome 37 (1987) no. 4, pp. 29-44. doi : 10.5802/aif.1109. http://www.numdam.org/articles/10.5802/aif.1109/

[1] A. Asada, Non-abelian Poincaré lemma, preprint, 1986. | MR | Zbl

[2] A. Besse, Einstein manifolds, Springer-Verlag, Berlin, 1987. | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

Cité par Sources :