On the natural operators on vector fields. (English) Zbl 0678.58003

Kolář, Ivan

Ann. Global Anal. Geom. 6, No. 2, 109-117 (1988).

Local product preserving functors from the category of manifolds into itself are given by the action of Weil-algebras A, by \(T_ A(M)=Hom(C^{\infty}(M),A)\). In this paper all natural (with respect to local diffeomorphisms of the base manifold M) lifts of vector fields on M to vector fields on \(T_ A(M)\) are determined. The answer is as follows. There is the flow prolongation: differentiate the local 1-parameter group functor applied to the flow. On this an arbitrary element of the Weil- algebra may act as generalized scalar multiplication. To this one may add one of the generalized Liouville vector fields, which are natural: they correspond exactly to the automorphisms of the Weil algebra.

Reviewer: P.Michor

MSC:

58A20	Jets in global analysis
53A55	Differential invariants (local theory), geometric objects

Keywords:

Weil-algebras; flow prolongation; group functor; generalized Liouville vector fields

References:

[1]	Doupovec, M.: Natural operators transforming vector fields to the r-th order tangent bundle. Časopis Pěst. Mat. (to appeear). · Zbl 0712.58003
[2]	Eck, D. J.: Product-preserving functors on smooth manifolds. J. Pure Appl. Algebra 42 (1986), 133–140. · Zbl 0615.57019 · doi:10.1016/0022-4049(86)90076-9
[3]	Ehresmann, C.: Introduction á la théorie des structures infinitésimales et des pseudo-groupes de Lie. Colloque du C.N.R.S., Strasbourg 1953, 97–110.
[4]	Gancarzewicz, J.: Liftings of functions and vector fields to natural bundles. Dissertationes Mathematicae, CCXII, Warszawa 1983. · Zbl 0517.53016
[5]	Kainz, G. and Michor, P.: Natural transformation in differential geometry. Czechoslovak Math.J. 37 (112) (1987), 584–607. · Zbl 0654.58001
[6]	Kolář, I.: Covariant approach to natural transformations of Weil functors. Comment. Math. Univ. Carolin. 7 (1986), 723–729. · Zbl 0603.58001
[7]	Kolář, I.: Some natural operations with connections. J. Nat. Acad. Math. India (to appear). · Zbl 0671.53023
[8]	Krupka, D., and Janyška, J.: Lectures on differential invariants. To appear. · Zbl 0752.53004
[9]	De León, M. and Rodrigues, P. R.: Almost tangent geometry and higher order mechanical systems. Differential Geometry and Its Applications, Proceedings, D. Reidel Publishing Company, 1987, 179–195. · Zbl 0637.58005
[10]	Luciano, O. O.: Categories of multiplicative functors and Morimoto’s conjecture. Preprint N{\(\deg\)} 46, Institut Fourier, Laboratoire de Mathématiques, Grenoble, 1986.
[11]	Morimoto, A.: Prolongation of connéctions to bundles of infinitely near points. J. Differential Geometry 11 (1976), 479–498. · Zbl 0358.53013
[12]	Nijenhuis, A.: Natural bundles and their general properties. Differential Geometry in honor of K. Yano, Kinokuniya, Tokyo 1972, 317–334.
[13]	Okassa, E.: Prolongements des champs de vecteurs á des variétés de points proches. C.R.A.S. Paris 300, (1985), série I, 173–176. · Zbl 0598.58003
[14]	Palais, R. S. and Terng, C. J.: Natural bundles have finite order. Topology 16 (1977), 271–277. · Zbl 0359.58004 · doi:10.1016/0040-9383(77)90008-8
[15]	Pohl, F. W.: Differential geometry of higher order. Topology 1 (1962), 169–211. · Zbl 0112.36605 · doi:10.1016/0040-9383(62)90103-9
[16]	Sekizawa, M.: Natural transformations of vector fields on manifolds to vector fields on tangent bundles. To appear. · Zbl 0657.53009
[17]	Weil, A.: Théorie des points proches sur les variétés différentiables. Colloque du C.N.R.S., Strasbourg 1953, 111–117.
[18]	Zajtz, A.: A method of estimation of the order of natural differential operators and bitings. To appear. · Zbl 0714.58059