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 , quand existe-t-il localement une connexion dont la courbure soit ? 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 , when does there exist locally a connection such that is the curvature of ? 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 -
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/
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