×

zbMATH — the first resource for mathematics

Projective connections. (English) Zbl 1114.53014
Two approaches (of È. Cartan and of T. Y. Thomas and J. H. C. Whitehead) to the study of projective connections are compared. Spray geometry is reviewed, the Thomas-Whitehead construction and its generalization to sprays are discussed, and afterwards the Cartan theory in the affine case is described. A certain bundle is introduced which realizes explicitly Cartan’s idea of attaching a projective space to each point of a manifold, called the Cartan bundle. This bundle is defined independent of any particular choice of connection, but to any projective class of sprays one can associate a unique Cartan connection on the Cartan bundle, as it is shown. In the course of the discussion a Cartan normal projective connection is derived for a system of second-order ordinary differential equations (extending the results of Cartan from a single equation to many) and the concept of a normal Thomas-Whitehead connection is generalized from affine to general sprays. Finally it is described in detail how to derive the Cartan connection from the generalized Thomas-Whitehead data for any projective equivalence class of sprays.

MSC:
53B10 Projective connections
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bailey, T.N.; Eastwood, M.G.; Gover, A.R., Thomas’s structure bundle for conformal, projective and related structures, Rocky mountain J. math., 24, 1191-1217, (1994) · Zbl 0828.53012
[2] A. C˘ap, Two constructions with parabolic geometries, 2005 (preprint), arXiv:math.DG/0504389
[3] Cartan, E., Sur LES variétés à connexion projective, Bull. soc. math. France, 52, 205-241, (1924) · JFM 50.0500.02
[4] Crampin, M., Connections of Berwald type, Publ. math. debrecen, 57, 455-473, (2000) · Zbl 0980.53031
[5] M. Crampin, Isotropic and R-flat sprays, Houston J. Math (in press) · Zbl 1125.53012
[6] Crampin, M.; Martínez, E.; Sarlet, W., Linear connections for systems of second-order ordinary differential equations, Ann. inst. H. Poincaré phys. theor., 65, 223-249, (1996) · Zbl 0912.58002
[7] M. Crampin, D.J. Saunders, On projective connections: the affine case (2004) unpublished. Available at http://maphyast.ugent.be Theoretical Mechanics · Zbl 1057.58008
[8] M. Crampin, D.J. Saunders, On projective connections: the general case (2004) unpublished. Available at http://maphyast.ugent.be Theoretical Mechanics · Zbl 1057.58008
[9] Doubrov, B.; Komrakov, B.; Morimoto, T., Equivalence of holonomic differential equations, Lobachevskii J. math., 3, 39-71, (1999) · Zbl 0937.37051
[10] Douglas, J., The general geometry of paths, Ann. math., 29, 143-168, (1928) · JFM 54.0757.06
[11] Fels, M.E., The equivalence problem for systems of second-order ordinary differential equations, Proc. London math. soc., 71, 221-240, (1995) · Zbl 0833.58031
[12] Fritelli, S.; Kozameh, C.; Newman, E.T., Differential geometry from differential equations, Comm. math. phys., 223, 383-408, (2001) · Zbl 1027.53080
[13] Grossman, D.A., Torsion-free path geometries and integrable second order ODE systems, Selecta math., 6, 399-442, (2000) · Zbl 0997.53013
[14] S. Hansoul, Existence of natural and projectively equivariant quantizations, 2006 (preprint), arXiv:math.DG/0601518
[15] Hebda, J.; Roberts, C., Examples of thomas – whitehead projective connections, Differential geom. appl., 8, 87-104, (1998) · Zbl 0897.53009
[16] Kobayashi, S., Theory of connections, Ann. di mat., 43, 119-194, (1957), p. 136 · Zbl 0124.37604
[17] Kobayashi, S.; Nomizu, K., Foundations of differential geometry, Vol. I, (1963), Interscience · Zbl 0119.37502
[18] Libermann, P., Cartan connections and momentum maps, (), 211-221 · Zbl 1029.53034
[19] Newman, E.T.; Nurowski, P., Projective connections associated with second order odes, Classical quantum gravity, 20, 2325-2335, (2003) · Zbl 1045.53013
[20] Nurowski, P.; Sparling, G.A.J., Three-dimensional cauchy – riemann structures and second-order ordinary differential equations, Classical quantum gravity, 20, 4995-5016, (2003) · Zbl 1051.32019
[21] Roberts, C., The projective connections of T.Y. Thomas and J.H.C. Whitehead applied to invariant connections, Differential geom. appl., 5, 237-255, (1995) · Zbl 0833.53023
[22] Schouten, J.A., Ricci-calculus, (1954), Springer, (Chapter VI) · Zbl 0057.37803
[23] Sharpe, R.W., Differential geometry: cartan’s generalization of klein’s erlangen program, (1997), Springer · Zbl 0876.53001
[24] Shen, Z., Differential geometry of spray and Finsler spaces, (2001), Kluwer · Zbl 1009.53004
[25] Thomas, T.Y., On the projective and equi-projective geometries of paths, Proc. natl. acad. sci., 11, 199-203, (1925) · JFM 51.0569.03
[26] Thomas, T.Y., A projective theory of affinely connected manifolds, Math. Z., 25, 723-733, (1926) · JFM 52.0732.02
[27] Whitehead, J.H.C., The representation of projective spaces, Ann. math., 32, 327-360, (1931) · Zbl 0002.15201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.