Higher order valued reduction theorems for classical connections

• Autorzy: Josef Janyška
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Natural bundle; natural operator; classical connection; reduction theorem; 53C05; 58A32; 58A20

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 2
• Strony: 294-308

Daty

wydano

2005-06-01

online

2005-06-01

Twórcy

autor

Josef Janyška

• Masaryk University

Bibliografia

• [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.
• [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.
• [5] I. Kolář, P.W. Michor and J. Slovák: Natural Operations in Differential Geometry, Springer-Verlag, 1993.
• [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.
• [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.
• [9] G. Lubczonok: “On reduction theorems”, Ann. Polon. Math., Vol. 26, (1972), pp. 125–133.
• [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
• [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.
• [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
• [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
• [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

Identyfikatory

DOI

10.2478/BF02479205