Previous |  Up |  Next


natural bundle; natural affine; vector bundle; natural differential operator; G-structure; structure tensor
The notion of a structure tensor of section of first order natural bundles with homogeneous standard fibre is introduced. Properties of the structure tensor operator are studied. The universal factorization property of the structure tensor operator is proved and used for classification of first order $*$-natural differential operators $\underline{D}:\underline{T\times T} \rightarrow \underline{T}$ for $n\ge 3$.
[1] Bernard, D.: Sur la gèometrie differèntiele des $G$-structures. Ann. Inst. Fourier 10 (1960). MR 0126800
[2] Epstein, D.B.A.: Natural tensors on Riemannian manifolds. J. Diff. Geom. 10 (1975), 631 -645. MR 0415531 | Zbl 0321.53039
[3] Fujimoto, A.: Theory of $G$-structures. Publications of the Study Group of Geometry 1 (1972), Okayama. MR 0348664 | Zbl 0235.53035
[4] Gancarzewicz, J.: Liftings of functions and vector fields to natural bundles. Diss. Math. CCXII (1983), Institute of Mathematics, Warszawa. MR 0697471
[5] Kolář, I., Michor, P.: All natural concomitants of vector valued differential forms. Proc. of the Winter School of Geometry and Physics (1987), Srní. MR 0946715
[6] Krupka, D.: Natural Lagrangian structures. Diff. Geom. 12 (1984), Banach center Publications, PWN - Polish Scientific Publisher, Warszaw, 185-210. MR 0961080 | Zbl 0572.58007
[7] Krupka, D., Mikolášová, V.: On the uniqueness of some differential invariants: $d,[,], \triangledown $. Cz. Math. J. 34(109) (1984), 588-597. MR 0764440
[8] Konderak, J.: Fibre bundles associated with fields of geometric objects and a structure tensor. preprint, International Centre For Theoretical Physics, Miramare, Trieste, 1987.
[9] Łubczonok, G.: On the reduction Theorems. Ann. Polon. Math. XXVI (1972), 125-133. MR 0307078
[10] Pommaret, J.F.: Systems of partial differential equations and Lie pseudogroups. Gordon and Breach, 1978. MR 0517402 | Zbl 0418.35028
[11] Terng, G.L.: Natural vector bundle and natural differential operators. Am. J. Math. 100 (1978), 775-828. MR 0509074
[12] Zajtz, A.: Foundations of differential geometry of natural bundles. Univ. of Caracas, 1984.
Partner of
EuDML logo