Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 082, 10 pages      arXiv:0908.1519      http://dx.doi.org/10.3842/SIGMA.2009.082
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Higher Order Connections

Michael G. Eastwood
Mathematical Sciences Institute, Australian National University, ACT 0200, Australia

Received March 12, 2009, in final form August 10, 2009; Published online August 11, 2009

Abstract
The purpose of this article is to present the theory of higher order connections on vector bundles from a viewpoint inspired by projective differential geometry.

Key words: connections; jets; projective differential geometry.

pdf (205 kb)   ps (149 kb)   tex (13 kb)

References

  1. Arnold D.N., Falk R.S., Winther R., Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1-155.
  2. Atiyah M.F., Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181-207.
  3. Branson T.P., Cap A., Eastwood M.G., Gover A.R., Prolongations of geometric overdetermined systems, Internat. J. Math. 17 (2006), 641-664, math.DG/0402100.
  4. Calabi E., On compact Riemannian manifolds with constant curvature I, in Differential Geometry, Proc. Sympos. Pure Math., Vol. III, Amer. Math. Soc., Providence, R.I., 1961, 155-180.
  5. Ciarlet P.G., An introduction to differential geometry with application to elasticity, J. Elasticity 78/79 (2005), no. 1-3, 1-215.
  6. Eastwood M.G., A complex from linear elasticity, in The Proceedings of the 19th Winter School "Geometry and Physics" (Srni, 1999), Rend. Circ. Mat. Palermo (2) Suppl. (2000), no. 63, 23-29.
  7. Eastwood M.G., Notes on projective differential geometry, in Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 41-60.
  8. Ehresmann C., Connexions d'ordre supérieur, in Atti Quinto Congr. Un. Mat. Ital. (Pavia-Torino, 1955), Edizioni Cremonese, 1956, 326-328.
  9. Goldschmidt H., Prolongations of linear partial differential equations. I. A conjecture of Élie Cartan, Ann. Sci. École Norm. Sup. (4) 1 (1968), 417-444.
  10. Jahnke P., Radloff I., Splitting jet sequences, Math. Res. Lett. 11 (2004), 345-354, math.AG/0210454.
  11. Libermann P., Sur la géométrie des prolongements des espaces fibrés vectoriels, Ann. Inst. Fourier (Grenoble) 14 (1964), 145-172.
  12. Olver P.J., Differential hyperforms I, Mathematics Report 82-101, University of Minnesota, 1982, available at http://www.math.umn.edu/~olver/a_/hyper.pdf.
  13. Spencer D.C., Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179-239.
  14. Virsik J., Non-holonomic connections on vector bundles. II, Czechoslovak Math. J. 17 (1967), 200-224.
  15. Yuen P.C., Higher order frames and linear connections, Cahiers Topologie Géom. Différentielle 12 (1971), 333-371.

Previous article   Next article   Contents of Volume 5 (2009)