Previous |  Up |  Next

Article

Keywords:
natural bundles; natural differential operators
Summary:
Let $F$ and $G$ be two natural bundles over $n$-manifolds. We prove that if $F$ is of type (I) and $G$ is of type (II), then any natural differential operator of $F$ into $G$ is of order 0. We give examples of natural bundles of type (I) or of type (II). As an application of the main theorem we determine all natural differential operators between some natural bundles.
References:
[1] J. Dębecki: Natural transformations of afinors into functions and afìnors. Supplemento ai Rendiconti del Circolo Matematico di Palermo (1991), in press.
[2] D. B. A. Epstein: Natural tensors on Riemmannian manifolds. Ј. Differential Geometry, 631-645. MR 0415531
[3] D. B. A. Epstein W.P. Thurston: Transformation groups and natural bundles. Proc. London Math. Soc. (3) 38 (1979), 219-236. MR 0531161
[4] J. Gancarzewicz: Differential Geometry. B.M.64, 1987. (In Polish.) MR 0944676
[5] J. Gancarzewicz: Liftings of functions and vector fields to natural bundles. Dissertationes Mathematicae, Warszawa (1982). MR 0663216 | Zbl 0503.53050
[6] T. Klein: Connection on higher order tangent bundles. Čas. Pӗst. Mat. 106 (1981), 414-424. MR 0637822
[7] I. Kolář: Functorial prolongations of Lie groups and their actions. Čas. Pěst. Mat. 108 (1983), 289-294. MR 0716414
[8] I. Kolář J. Slovák: On the geometric functors on manifolds. Suplemento ai Rendiconti del Circolo Matematico di Palermo (1988), 223-233. MR 1009575
[9] W. M. Mikulski: Continuity of liftings. Čas. Pěst. Mat. 113 (4) (1988), 359-362. MR 0981877 | Zbl 0677.58006
[10] A. Morimoto: Prolongations of connections to bundles of infinitely near points. Ј. Differential Geometry 11 (1976), 479-498. MR 0445422
[11] A. Nijenhuis: Natural bundles and their general properties. Differential Geometry, Kinokuniya, Тokio (1972), 317-343. MR 0380862 | Zbl 0246.53018
[12] R. S. Palais C. L. Terng: Natural bundles have finite order. Тopology 16 (1977), 271-277. MR 0467787
[13] J. Slovák: On the finite ordeг of some operators. Proc. of the Conf., Brno, 1986.
[14] A. Zajtz: Foundation of differential geometry on natural bundles. Caracas, 1984, preprint.
Partner of
EuDML logo