Connections with prescribed curvature

Dennis Deturck; Janet Talvacchia

Annales de l'institut Fourier (1987)

  • Volume: 37, Issue: 4, page 29-44
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Deturck, Dennis, and Talvacchia, Janet. "Connections with prescribed curvature." Annales de l'institut Fourier 37.4 (1987): 29-44. <http://eudml.org/doc/74779>.

@article{Deturck1987,
abstract = {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.},
author = {Deturck, Dennis, Talvacchia, Janet},
journal = {Annales de l'institut Fourier},
keywords = {prescribed curvature; connection; principal bundle; nilpotent structure groups},
language = {eng},
number = {4},
pages = {29-44},
publisher = {Association des Annales de l'Institut Fourier},
title = {Connections with prescribed curvature},
url = {http://eudml.org/doc/74779},
volume = {37},
year = {1987},
}

TY - JOUR
AU - Deturck, Dennis
AU - Talvacchia, Janet
TI - Connections with prescribed curvature
JO - Annales de l'institut Fourier
PY - 1987
PB - Association des Annales de l'Institut Fourier
VL - 37
IS - 4
SP - 29
EP - 44
AB - 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.
LA - eng
KW - prescribed curvature; connection; principal bundle; nilpotent structure groups
UR - http://eudml.org/doc/74779
ER -

References

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  1. [1] A. ASADA, Non-abelian Poincaré lemma, preprint, 1986. Zbl0607.58002MR88d:58005
  2. [2] A. BESSE, Einstein manifolds, Springer-Verlag, Berlin, 1987. Zbl0613.53001MR88f:53087
  3. [3] R. BRYANT, S. S. CHERN, R. GARDNER, H. GOLDSCHMIDT and P. GRIFFITHS, Exterior differential systems, to appear. Zbl0726.58002
  4. [4] E. CARTAN, Les systèmes différentiels extérieurs et leurs applications géométriques, Paris, Hermann, 1945. Zbl0063.00734MR7,520d
  5. [5] D. DETURCK, Existence of metrics with prescribed Ricci curvature : Local theory, Inventiones Math., 65 (1981), 179-207. Zbl0489.53014MR83b:53019
  6. [6] D. DETURCK and D. YANG, Local existence of smooth metrics with prescribed curvature, Contemporary Math., 51 (1986), 37-43. Zbl0587.53040MR87j:53059
  7. [7] J. GASQUI, Sur la résolubilité locale des équations d'Einstein, Compositio Math., 47 (1982), 43-69. Zbl0515.53015MR84f:58115
  8. [8] J. GASQUI, Sur l'existence locale d'immersions à courbure scalaire donnée, Math. Ann., 219 (1976), 123-126. Zbl0303.53025MR52 #12013
  9. [9] H. GOLDSCHMIDT, Existence theorems for analytic linear partial differential equations, Annals of Math., 86 (1967), 246-270. Zbl0154.35103MR36 #2933
  10. [10] H. GOLDSCHMIDT, Integrability criteria for systems of non-linear partial differential equations, J. Diff. Geom., 1 (1967), 269-307. Zbl0159.14101MR37 #1746
  11. [11] V. GUILLEMIN and S. STERNBERG, An algebraic model of transitive differential geometry, Bulletin Amer. Math. Soc., 70 (1967), 16-47. Zbl0121.38801MR30 #533
  12. [12] M. KURANISHI, Lectures on involutive systems of partial differential equations, Lecture notes, Soc. Math. Sao Paulo, 1967. Zbl0163.12001
  13. [13] B. MALGRANGE, Équations de Lie II, J. Diff. Geom., 7 (1972), 117-141. Zbl0264.58009MR48 #5128
  14. [14] J. TALVACCHIA, Ph. D Thesis, U. of Pennsylvania. 
  15. [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). Zbl0516.53014MR84e:53043
  16. [16] D. ZHELOBENKO, Compact Lie groups and their representations, A.M.S. Translations of Math. Monographs, 40 (1973). Zbl0272.22006

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