We generalize reduction theorems for classical connections to operators with values in k-th order natural bundles. Using the 2nd order valued reduction theorems we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.
@article{bwmeta1.element.doi-10_2478_BF02479205,
author = {Josef Jany\v ska},
title = {Higher order valued reduction theorems for classical connections},
journal = {Open Mathematics},
volume = {3},
year = {2005},
pages = {294-308},
zbl = {1114.53018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02479205}
}
Josef Janyška. Higher order valued reduction theorems for classical connections. Open Mathematics, Tome 3 (2005) pp. 294-308. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02479205/
[1] E. B. Christoffel: “Ueber die Transformation der homogenen Differentialausdücke zweiten Grades”, Journal für die reine und angewandte Mathematik, Crelles's Journals, Vol. 70, (1869), pp. 46–70. http://dx.doi.org/10.1515/crll.1869.70.46
[2] J. Janyška: “Natural symplectic structures on the tangent bundle of a space-time”, In: Proc. of the 15th Winter School Geometry and Physics, Srní (Czech Republic), 1995; Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, Vol. 43, (1996), pp. 153–162.
[3] J. Janyška: “Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold”, Arch. Math. (Brno), Vol. 37, (2001), pp. 143–160. | Zbl 1090.58007
[4] J. Janyška: “On the curvature of tensor product connections and covariant differentials”, In: Proc. of the 23rd Winter School Geometry and Physics, Srní (Czech Republic) 2003; Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, Vol. 72, (2004), pp. 135–143. | Zbl 1051.53017
[5] I. Kolář, P.W. Michor and J. Slovák: Natural Operations in Differential Geometry, Springer-Verlag, 1993. | Zbl 0782.53013
[6] O. Kowalski and M. Sekizawa: “Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles-a classification”, Bull. Tokyo Gakugei Univ., Sect. IV, Vol. 40, (1988), pp. 1–29. | Zbl 0656.53021
[7] D. Krupka: “Local invariants of a linear connection”, In: Colloq. Math. Societatis János Bolyai, 31. Diff. Geom., Budapest 1979, North Holland, 1982, pp. 349–369.
[8] D. Krupka and J. Janyška: Lectures on Differential Invariants, Folia Fac. Sci. Nat. Univ. Purkynianae Brunensis, Brno, 1990. | Zbl 0752.53004
[9] G. Lubczonok: “On reduction theorems”, Ann. Polon. Math., Vol. 26, (1972), pp. 125–133. | Zbl 0244.53011
[10] A. Nijenhuis: “Natural bundles and their general properties”, Diff. Geom., in honour of K. Yano, Kinokuniya, Tokyo, 1972, pp. 317–334.
[11] G. Ricci and T. Levi Civita: “Méthodes de calcul différentiel absolu et leurs applications”, Math. Ann., Vol. 54, (1901), pp. 125–201. http://dx.doi.org/10.1007/BF01454201 | Zbl 31.0297.01
[12] J.A. Schouten: Ricci calculus, Berlin-Göttingen, 1954.
[13] M. Sekizawa: “Natural transformations of affine connections of manifolds to metrics on cotangent bundles”, In: Proc. of the 14th Winter School on Abstract Analysis, Srní (Czech Republic), 1986; Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, Vol. 14, (1987), pp. 129–142.
[14] M. Sekizawa: “Natural transformations of vector fields on manifolds to vector fields on tangent bundles”, Tsukuba J. Math., Vol. 12, (1988), pp. 115–128. | Zbl 0657.53009
[15] C.L. Terng: “Natural vector bundles and natural differential operators”, Am. J. Math., Vol. 100, (1978), pp. 775–828. http://dx.doi.org/10.2307/2373910 | Zbl 0422.58001
[16] T.Y. Thomas and A.D. Michal: “Differential invariants of affinely connected manifolds”, Ann. Math., Vol. 28, (1927), pp. 196–236. http://dx.doi.org/10.2307/1968367 | Zbl 53.0684.02
[17] R. Utiyama: “Invariant theoretical interpretation of interaction”, Phys. Rev., Vol. 101, (1956), pp. 1597–1607. http://dx.doi.org/10.1103/PhysRev.101.1597 | Zbl 0070.22102